COMBmATORIAL PROBLEm niTRE
THEORY OF FACTORIAL DESIGNS AND ERROR CORRECTnc CODES
by
Pierre Robillard
University of North Carolina
Institute of Statistics Mimeo Series No. 594
October 1968
This research was supported by the U. S. ArrIr3
Research Office-Durham under Grant Number
DA-ARO-D-3l-J24-G9l0
DEPARiH:NT OF STATISTICS
University of North Carolina
Chapel Hill, N. C. 27514
TABLE OF CONTENTS
gwpt.r
Title
I
II
i
Table of Contents
11
Aoknowledgements
iv
SUJIDI18.ry
v-x
THE THEORY OF SlMME'l'RICA.L FACTORIAL DESIGN
1.
Factorial designs
1
2.
Det1.nitions and notations
2
3.
Confounded factorial designs
6
4.
Confounded designs of the class
5.
Modular representation of the (sn,sk)
confounded factorial designs
13
6.
Necessar,y conditions for equivalent designs
18
(,p, sk)
8
ENUMERATION OF (sn, sk) CONFOUNDED FACTORIAL DESIGNS
1.
Technique of enumeration
21'
2.
Construction of (sn, sk) confounded designs
for the case k = 2,3,4
25
3.
Types of equivalent designs
26
4.
The n~ber of different types of equivalent
(sn, s ) designs.
,
The enumeration of the types of (2n ,23 )
confounded design
5.
28
30
•
iii
In
"
~
IV
\........
CONNECTIONS BETWEEN THE TJm)RY OF OON1l'OUNDED FACTORIAL
DESIGNS AND THE THEORY OF oomm
1.
The theory of coding - Introduction and' defirdtions
65
2.
Additive errors
66
3.
Codes with algebraic structure - Linear codes
67
4.
Linear codes and factorial designs
70
5.
Definitions of optimal1ty
73
6.
Optimality in the case s=2, k=3
80
7.
The dimensions of the sets
c.
83
WEIGHT mSTRIBUTION OF LINEAR CODES
1. Introduction
84
2.
Notations and definitions
85
3.
Second formula for the spectrum
89
4.
Bounds on A( v)
95
5.
Future research
'99
BIBLIOGRAPHY
100
APPENllIX I
102
APPENmX II
106
ACXNOWLEOOEMENTS
To the members of the faculty of the Department ot
statistics who have contributed to my graduate training, I wish
to express my sincere gratitude.
Special thanks are owed to ay
adviser, Protessor I. M. Chakravarti, ror his guidance and encourage.
ment during my research.
I would like to thank Proressors R. C. Bose, T• .1. Dowling,
J. E. Grizzle, N. L. Johnson, R. J. Monroe, the other members ot .y
doctoral committee, for their review of my work and their suggestions
for improving it.
I am indebted to the Department of Statistics, the
Minist~re de 1 'Education de la province de Quebec, the U. S. Army
Research Oftice. Durham, for providing financial support for .y
graduate study.
appreciation.
To these institutions I gratefully express my
//
SUMMARY
The factorial experiment is a general technique for the
study of the variations brought about by deliberate changes in the
experimental conditions.
In general, experiments are carried out
by scientists in order to determine the effects of one or more factors
on the yield or quailty of a product or the measure of an instrument
or the resistance of a material, etc. • ••
If the experiment is
such that the conclusions can be drawn about each factor independently,
a great advantage of simplicity in the interpretation is gained.
To
achieve this objective one can decide on the set of values or levels
for each of the factors to be studied and carry out one or more
trials of the process with each of the possible combinations of the
levels of the factors.
me nt.
We call
such experiment a factorial experi-
We use the term factor to denote any feature of the experi-
mental conditions which will be controlled by the experimenter.
The various values of a factor examined in the experiment are called
the levels.
The combination of levels of all factors used at a given
trial is known as the treatment combination and the result of such
a trial is the response.
Because we want to compare different treatments together and
be able to tell how they differ we need to perfonn the experiment
under conditions that are nearly as alike as possible for each
treatment.
Considerable variations in the experimental units to
vi
which the treatments are applied would vitiate the conclusions to be
drawn.
An efficient way to overcome this difficulty when variations
are known to be present in the experiment is to perfom what is known
as blocking or confounding.
One divides the experimental material
(which may be the field where the crop grows, the pieces of machinery,
etc•••• ) into blocks or subsets in which the experimental conditions
are much alike.
In each block a fraction of the total experiment
is perfomed.
In many cases however it is possible to assign the
treatments to the different blocks in such a way that we are able on
one hand to free the more inportant comparisons among treatments
from the relatively larger variations between blocks while on the
other hand we deliberately confound unimportant comparisons with
these variations.
These methods of assigning treatments to blocks
is part of the design of an experiment and it is this aspect we want
to study and we will specialize to the case of the symmetrical
factorial experiments.
We consider a sn symmetrical factorial experiment
involving n factors, denoted by F .F ••••• F • each at s levels.
1 2
n
s a prime number or power of a prime.
factors occur at levels
T(xl'x2 " " ,~),
~ ,~ ....
A treatment in which these
,x ' respectively, is denoted by
n
Following Bose [2] we identify the s levels at which
a factor occurs with the elements or a Galois field GF(s).
The
treatment T(x1'~"" ,xn ) may be identified with the point
(xl'x2 .... ,xn ) of the n-dimensional finite Euclidian space men,s).
A 1:1 correspondence is thus established between the sn treatments
and the sn points of the space men,s).
Splitting and confounding
/'
/
vii
of degrees of freedom can be translated in geometrical tel'll1s.
A symmetrical factorial design on sn treatments is said
to be of class (sn, sk) if each replication is laid out in sk blocks
(k 'n) of sn.k treatments each.
In suoh a design, (sk.1 ) degrees
of freedom are confounded with blocks.
The point of interest for the
experimenter is the choice of the etfeots which will be oontoumed.
It is known that one does not have OOIIlplete liberty in the ohoice
of confounded effects: in faot, one can ohoose exactly k of the
effects for confounding before the total .cheme of oontouming in the
experiment is completely detel'lllined.
'!he 1II&1.n proble is thus
to be able to enU1llerate all the (sn ,sk) designs and their respective
confounding schemes.
Connected to this is the problem ot detel'lllining
the maximum number of factors when interactions up to a given order
are to be left unconfounded.
'!hese problems were studied in (2] •
where it was shown how the (sk.1 ) confounded degrees of freedom are
related to the k generating factors an experimenter may choose to
detel'llline the design.
A method of approach to the problem of
enumeration of the (sn, sk) confounded designs is described.
Based on the treatment of a factorial experillent developed
in [2), we describe in this work a new geometrical method of
enUmeration which is relatively easy to apply in the case of
(sn,s2), (sn,s3) and (sn,s4) designs.· In Chapter I, which serves as
tis method uses the modular representation of the matrix generating
the confounding scheme ot the design, the colwms of the generating
matrix being regarded as points in a projective g8Qllletry.
We study.
1hen the different kinds of designs associated with different conf'igurations of the points associated with the generating matrix
am
we giva necessary conditions for the existence and feasibility or
certain designs.
We have worked out the cOll1plete descriptions ot the
6 3
6 4
designs of class (s •s ) and ~ • s ).
in the Appendix.
These are given in the tables
We have also solved oOll1pletely the case (sn. s2) and
n
(2 .2 3 ).
One of the interesting features of the mathematical theory
of factorial designs is its olose connection with the relatively uw
theory of error-ooding and the fact that the problems of coding can
be often tre.nslated into design problems and vice versa.
But let us
first introduce briefly the theory of coding.
The theory of ooding consists of an ensemble of techniques
and methods for aocurately tranS1l1itting signals over noisy ohannels.
We shall consider the situation of disorete messages or signals
produoed by a source
am
encoded into a sequence of symbols which
are sucoessively sent over the channel (e.g. a telephone line. the
interstellar space. etc•••• ).
Beoause of the presence of some noise
or disturbances during the transmission. the received symbols ma.y
not correspond to the original transmitted symbols.
At the receiver
end it is possible to recover some or the information lost during the
transmission by a decoding operation.
a code.
This is done with the aid ot
We shall restrict our attention in this work to block codes
in which a message correspond to a block of k symbols; each block or
k sYlllbols produced by the source and made available at the encoder
is transformed into a block of n symbols called code word which is
/
ix
sent over
t~e
channel.
The set of code words is called a code.
At t.!-}e receiver end, Ule received block of n-symbols may
contain some error but because the code does consist of a subset
of all possible n-symbols the erratic sequence is not necessarily
identified as a code work; the detection or even correction of the
error is often possible.
The main problem is then the proper selection
of n-symbols to fom the oode.
We shall aSSUMe that the channel is able to transmit
s-distinct symbols, s a prime or power of a prime.
We identify these
s symbols with the elements of a Galois field GF(s).
A sequence of
n symbols is identified with an n-vector of V the vector space of
n
dimension n over GF(s).
A code is a sUbset of V.
n
If this subset
is itself a linear subspace of dimension k we talk of an (n,k)
--
,
linear s-ary code.
We define the weight of a code word as the number
of its non null coefficient.
The notion of weight and error
correction capability of a code is related as follows:
in a linear
code with llin1mum word-weight equal to (2t + d + 1) it is possible to
correct up to t error and detect up to t + d
errors.
In Chapter III we shall present in a mathematical languaC.
the notion of a code and how we can relate it to the notion of a
confounded factorial design.
Speaking very briefly one can identify
to every (n,k) linear s-ary code a (sn, sk) confounded factorial
design such that a code word of weight w corresponds with a (w-1)order confounded interaction.
\ie show how the problem of designs can
be translated in coding language.
We shall conclude this chapter
by stressing the cormections between both topics studied above, factorial designs and theory of codes, with the theory of fractional
factorial designs.
In Chapter IV in the context of coding theory. we
shall investigate the problem of the weight distribution of a linear
code.
In the utilization of a code in a cOlllll1um.cation system an
important parameter to consider is the spectrum of a code,that
is the distribution of the code words by weight.
Our interest will
be limited in this work to t."le aspect of deriving the weight distribution of a code.
We shall start by a derivation of a general
fonnula for weight distribution in linear code based on the Mobius
inversion fonnula.
We deduce the known formula of weight distribution
for optimum code and show how this can be utilized to obtain necessary
conditions for the existence of
SUCh
codes.
In the second stage
we shall transfonn the original fonnula into one giving the weight
distribution as a function of quantities as N~, the number of sets
of u columns of the parity check matrix of the code which span a
v dimensional space.
This is used to derive a new formula for the
conic code and distance 3 s-ary Hamming code.
quency is
tabul~ted
4 s-ary conic
for s
code.
=
A table of weight fre-
3,4,5,7,8 in the case of distance
Finally using the last fonnula for weight
distribution of a code we derive a lower and upper bound on the
weight distribution of a code and compare it with previous ones.
/
/
C!iAPTER I
TIIE TIIEORY OF SYMMETRICAL FACroRIAL DESIGNS
1 - Factorial designs
Many experimental situations require the study of the
effect of varrying two or more factors.
In the exploration of such
situations one knows that it is not sufficient to examine the
variation of one factor 'at a time but all combinations of the different
factor levels must be examined in the same experimental set up in
order to explain the effect of each factor and the possible way in
which each factor may be modified when others are changed.
Sucn a procedure called factorial experiment has many
advantages over the one that examines one factor at a time.
estimates each effect with the same accuracy as
had been varied at a time.
It first
if only one factor
It also gives an estimate for each
interaction between the factors, this being essential if the research
wOrker wants to acquire sCllle insight into the phenomena under study.
When the number of factors is large the dimension of the
factorial experiment is large also and it may be difficult in that
case to perform the experiment in homogeneous way; the variations
of the experimental material becOllle important and can vitiate the
conclusions of the experiment and reduce considerably the efficiency
of the experimental set up.
For example, a large biochemical
2
experiment on rats may require a large number of th. to receive
the treatments,but large variations in the condition of the rats
fran 11tter to 11tter may reduce the efficiency of the exper:1aent
and obscure the real difference between the treatments.
The sise of
a large industrial experiment may require that the experimental
material be taken from various locations which brings in external
variations in the experiment which may weaken the results.
'!he most efficient way of dealing 'With this difficulty
is to d1vide the experimental material in SIlaller hanogeneous blocks
and assign subsets of the treatments to each block in such a way that
it is now possible to estimate some effects with very great precision
as if the external variations were not existing.
A major inconvenience in such a procedure is that sane
other effects are confounded; that is the variation between blocks
will obscure any statement about them.
It is often possible however
.
to select these confounded effects as being the ones that are known
to be unimportant or non-existent and lose very little in the interpretation of the results.
In order to do that in various situations
we must know the different schemes
eaoh kind of designs.
of confounding available for
It is the purpose of this work to present
a method of constructing confounded designs and describe their
confounding schemes.
We first introduce some notations and defi-
nitions.
2 - Definitions
~
notations
Consider a sn symmetrical factorial design involving n
factors F ,F , ••• ,F each at s levels, s a prime number or power
1 2
n
:3
of a prime number.
The effect of a treatment in which these factors
occur at levels X1,x2 , ••• ,xn ' respectively. is denoted by T(x1 ,x2 •••• ,
x ). A. linear function of the treatment effects may be written in
n
the fom
L =
Xi taking all the s possible values.
A linear function for which
r
is called a contrast.
There are (sn_1) such independent contrasts
in the space of linear functions of treatment effects and the set
of all contrasts forms a vector space of dimension (sn_ i ).
say that
eac~
We will
contrast possesses a degree of freedom and in that
sense we shall talk of a degree of freedom belonging to a contrast.
The degrees of freedom belonging to two orthogonal contrasts will
be defined as orthogonal.
A set of contrasts containing r independent
contrasts is said to possess r degrees of freedom.
n
There are alto-
.
gether (s -1) degrees of freedom.
The contrast between the two set. of e different treatment
2 - •.. - T~.
••• + Te - Ti - T
A. contrast is said to belong to the main effect Fi if the
coefficients in the linear function depend only on the levels of the
factor F ; i.e•• we can write the contrast as
i
4
A contrast is said to belong to the first order interaction
Fi F j between the factors Fi and F j if the coefficients of the func-
tion depend only on the levels of the factors Fi and F j and the
contrast is orthogonal to any contrast belonging to the main effect
of F or F •
j
i
One can write such a contrast as
wi th the property that
1:
= O.
A contrast is said to belong to the r-th order interaction
of the factors Fi ' Fi , ••• , Fi
1
2
if the coefficients of the function
r+1
depend only on the levels of the factors F ,F , ... ,F
and
i1 i2
i r+1
the contrast is orthogonal to the contrasts belonging to any main
effect of these factors and any k-th order interaction of these
factors for k = 1, 2, ••• , r-1 and r Ln.
A r-th order interaction
is also called a (r+1 )_. factor interaction.
It is easily seen that
t.~e
contrasts belonging to a r+1
factor interaction Fi ,Fi , ••• , Fi
form a (s1"+1_1) dimensional vec1xr
1
2
r+1
subspace in t.~e space of contrasts. The spaces of contrasts belonging
to different interactions being orthogonal, the whole (sn_ 1 )__ space of
contrasts can be split into
n
(~) + (~) + ••• + (~) = 2 _1
orthogonal subspaces and thus the sn_ 1 degrees of freedom are split
into 2n .1 corresponding orthogonal sets of degrees of freedom.
We shall now identify the s levels of each factor with the
c::
..1
elements of' a Galois field GF( s) and
(x1 .X2 ••••• X ) with the points P
n
~clidian
geometry EG(n. s).
between the s
n
t~e
treatment cO!ll.bir.a.tions
= (x1.~ ••••• xn)
of a finite
'Ihis establishes a 1:1 correspondence
treat.'I1ents of the experiments and the s
of the geometry EG(n.s).
n
points
Such correspondence was first established
by Bose in [2].
A parallel pencil P(a .a ••••• a } in EG(n.s) consists of
n
1 2
s parallel In...1 )......flats.
Each (n-1 )--fiat is defined as the set of
points (xl'~'." .xn ) satisfying the equation
a 1 Xl + a 2 x 2 + ••• + an x n
= ao '
a i £ GF(s).
The s parallel (n-l )--flats of th.e pencil are obtained
by letting a
o
take the s values of GF( s).
We shall say that two
parallel pencils P(a .a ••••• a ) and P(b ,b , •••• b ) are identical
1 2
1 2
n
n
when a
i
=
s> bi
for i = 1, 2, ••• , n and
9 .,
O,an element of GF(s).
The s parallel flats of t.lote pencils divide the sn points of m{n.s)
into s disjoint sets of sn-l points each.
Speaking in tems of the
treatments, one has split the sn treatments into s disjoint sets of
sn-l treatments each.
The contrasts between these s disjoint sets
carry (s-O degrees of freedom which are said to belong to the pencil.
It can be shown that t.lote sets of (s-1) degrees of freedom
carried by two different pencils are ort.lotogonal.
This and the-fact
t.~at there are (sn_ 1 )/(s_1) different parallel pencils in EG(n,s) show
that one can divide the (sn_ 1) degrees of freedom into (sn_1 )/(s-1)
orthogonal sets of (s-l) degrees of freedom each.
Moreover it can
be shown that t."le nature of the degrees of freedom carried or eon",
founded by a given parallel pencil P(al'a2 ••••• a n ) depends on its
coordinates a ,a •••• ,an'
1 2
Precisely if just k coordinates or the
6
then the (5-1)
negr~es
ai ,a , ••• a
i
1i 2 k
of freedom carried by the pencil belong to the
k-factor interaction F F ••• Fi (a 1-factor interaction is a main
i1
i 2k
effect); in such case we shall say that the pencil belongs to the
k-factor interaction Fi Fi
••• Fi •
12k
In summary one can divide the total set of (sn_1) degrees
of freedom carried by an experiment into (sn_ l )!(s_l) orthogonal sets
of (s-1) degrees of freecom each, all the degrees of freedom in one
set belonging to the same interaction, m.d a given k-factor interaction
possesses (s_l)k-l of these sets of (s-1) degrees of freedom.
'3 - Confounded factorial,designs
Let us consider a parallel pencil P(a t ,a , ••• ,a ) which
l
n
.
t
splits the entire set of treatments into s disJoint subse s of s n-l
treatments each.
If each subset of treatments is assigned to a block
the design obtained is
sai~
to confound the interaction corresponding
to P(a 1 ,a2 •••• ,an ). In this case any contrast among the subsets
defined by the pencil cannot be estimated free from the block effect.
4
For example in the case of a, experiment where the
treatments are represented by the points (x l ,x2 'Xj,x4) of EG(4,J),
xi £ GF(J), one may consider the parallel pencil P(1,O,1,2); the.
treatments are then assigned to the blooks as follows:
the set
the set
{ (x ,x ,x ,x )
1 2 3 4
{ (x1 ,x2 ,x) ,x4 )
the set
{ (x ,x ,x ,x )
l 2 3 4
(all the operations are mod. 3).
xl + x + 2x4 -= O} goes in block 1,
3
x 1 + x + 2x4 = 1} goes in block 2,
3
xl + x 3 + 2x4 = 2} goes in block 3.
The two degrees of freedom carried by P(1.0,1,2) became
7
confounded in such an experiment; these two degrees of freenom belong
to the interaction F
1 F3 F4• In this experiment all the contrasts
not carrieo by P(l,O,l,2) can be estimated free from any block
effect.
We now proceed to confound the degrees of freedom
belongin~
to two different parallel pencils P(a ,a2 , ••• ,a ) and P(b ,b , ••• ,b ),
1
1 2
n
n
say. We consider the set of treatments given by
(Xl ,~, •• , ,xn )
I
a 1x 1 +
a2~
+ ••• + anxn = a O }
=
{
b1x 1 + b2x2 + ••• + bnxn bO
This set is a (n-2)--flat of EG(n,s) intersection of the (n-a)--
flat belonging to P(a 1 ,a2 , ••• ,a ) defined by
n
I a1x 1 + a2~+ ••• anxn = ao}
{ (x1'x2 '··· ,xn'>
and the (n-l)--flat belonging to P(b1 ,b2 , ••• ,b ) defined by
n
{ (Xl ,x2 '·· .,xn )
I b1x1 + b2x2 + ...
+ bnxn = bol •
The set of s2 disjoint (n-2)--fiats obtained by letting
a O' b O take all the values of GF(s) is called a parallel bundle of
order 2. Such bundle splits the set of treatments into s2 disjoints
subsets of s n-2 treatments each.
If we assign each subset to a block
the design obtained is known to confound the (s-l) degrees of freedom
belonging to P(a 1 ,a2 , ••• ,a ) and P(b1 ,b , ••• ,b ). Moreover such
n
2
n
design confounds the (s-l) degrees of freedom carried by
P(~lal +~2bl' .A 1a 2 + ~la2 + ~2b2'·"').lan + ~2bn) where
~ l' }. 2 E. GF( s) and (). l' A2)
f.: ( 0,0) •
In the example above if we choose for second pencil
2
P(O,2,l,2). the 3 sets defined are
{
(Xl ,x2,~,x4)
I
~
~
+ 2x4 = a O }
+ x3 + 2x4 b O
xl +
=
8
The confounded pencils in this case are
when a O' b O + O. 1. 2.
P(1.0.1.2)
belonging to F1 F:3 F4
P(0.2.1.2)
belonging to F2 F:3 F4
P(1.2.2.1)
belonging to F1 F2 Fj '4
PO.1.0.0)
belonging to F1 F2
each carrying two degrees of freedom.
In general the degrees of freedom earried by k different.
pencils P1 • P ••••• P can be simultaneously oonfounded by assigning
2
k
each of the sk sets of the parallel bundle of order k defined by these
to one block.
'Ihe totality of the sk-1 degrees of freedom oonfounded
in this way is carried by the (sk_1 )!(s_1) pencils given by
peA
1a 11 + ~2a12 + ••• + ). k a 1k:· .. ·' ~1an1 + ) 2an2 + ...
+~kank)
where
Pj
Xi
takes all the possible values of GF(s) i
= P(a1j .a2j ••• .,anj ).
j
= 1.2 ••••• k.
= 1.....k
and
(we do not allow all the ~ i
to vani sh simultaneously).
4 - ConfQundgd designs .oJ: the class 'sn. sk)
A symmetrical factorial design sn is said to be of class
(sn, sk) if each replication is laid out in sk blocks (k~ n) ot
s n-k treatments each.
We have just seen that a way to look at the
n
k
.
problem of construction of an (s • s ) design is to consider k
independent parallel pencils Pi' P2' •••• Pk in men,s) with
9
P1 = P(a11·a12·····a1n)
P2 = P(a21·a22·····a2n)
•
•
•
•
• aijE GF(s)
= «aij »
A
has rank k in GF(s)
Pk = P(~1·~2·····~n)
and think of Pi as the main effect pencil of general factor
ii(i
= 1.2 ••••• k).
The intersection of the k pencils P1 .P2 •••• 'Pk defines
a partition of m(n.s) in sk disjoint sets of sn-k points each and
correspondingly a partition of the s
treatments.
n
treatments in s
k
sets of s
n-k
If each set of treatments is assigned to a block. the
design obtained is of class (sn. sk).
We shall say that Pl'P2 •• .. 'Pk
are the generating pencils of the design.
a design each of the general factors
constant level in each of the block.
It is clear that in such
t l' t 2' •••• t
k remains at
k
Thus the s -1 degrees of freedClll
belonging to all possible generalized interactions of i l'
t 2'· • • • t
k
are confounded.
In terms of the coordinates of the generating pencils the
(sk_1 ) degrees of freedom confounded belong to ·the (sk_1 )/(s_1)
parallel pencils P(b .b2 ••••• b ) where the coordinates (b1 .b2 •••••b n )
1
n
are given by
(bl'b2 • .. • .bn ) = (,\ l' A2···· • ~ k)
(4.1)
'\1'\2 ••• '\n
10
It is understood that two vectors (b1 .b2 ••••• b ) and
n
(bt. b
i
2..... b~)
are the coordinates of the same pencil if bi
=J bi
= 1.2 ••••• n.
The set of vectors (b l .b2 ••••• b ) defined by equation
n
( 4. 1) completely define the confounding structure of the design.
Depending on the positions of the zero coordinates in each vector
(b 1 .b2 ••••• b n ),the corresponding pencil will carry degrees or freedom
belonging to a specific interaction.
Let us denote the set or
(sk_1 )/(s_1) vectors defined by (4.1) by L(A) where A = «a » is
ij
called the generating matrix of the design.
Let us define b~ =
then the z-vector
( b Z .b Z ••••
1
2
1 if bi
'=
o if bi
=0 :
0
,b z) indicates where are the zero
n
aoordinates of the corresponding vector (b , b , ••••b ).
l 2
n
Wi th this
notation we can say the the (s-1) degrees of freedom confounded by
a pencil P(b .b ••••• b ) belong to the interaction
l 2
n
Z Z
Z
b
b
b
l
2
n (in the Yates' notation).
F
l F2 ... Fn
For example continuing the illustration of Section 3
and considering (b l ,b .b .b 4 )
2 3
(~~.b~,b~.b~) = (0,1.1,1).
o
= (0.2,1,2)
we have that
and the two degrees of freedom confounded
1
1
1
by P(O,2,l.2) belong to F1 F2 F) F4
= F2
F) F4 •
We now define the symbolic notation
,..z
and
bZ
1
the set L!(A)
= Fl
bZ
2
bZ
n
F2 ••• Fn where I' = (F1 .F2 ••••• Fn )
we let the set L(A) ={h'
R' = ~'A: b. i: Q} correspond to
z
II
.l:'
={~
I
R' e
L(A)} which consists of the ftrious
interactions to which the (sk-1) / ( s-1) sets or (s-1) degrees of
11
freedom confounded belong.
Let us consider the confounded design D of class (sn. sk)
1
generated by the k independent pencils P • P1Z ••••• Plk and with
1l
corresponding generating matrix Al , and a second confounded design )2
of class (sn, sk) generated by the k independent pencils P ' P '
Z1
ZZ
• • •• P2k and with corresponding generating matrix AZ• Then we will
say that 01 and 0z are equivalent if the set of confoumed interactions LI (A l ) can be obtained from the set t.I(AZ) by changing the
name of the factors. or in other words if there exists a n x n
pemutations matrix E such that
n
(4.2)
In the:3
4
.
LI(At )
E F
= L n-(AZ).
factorial experiment described previously the
two pencils P(O.Z,l.Z) and P(1,0.1.Z) generate a ()4, )Z) confounded
factorial design 01; the generating matrix is
A
1
the set L(A 1 ) =
=
t (OZ1Z).
10
2 1 2]
101 Z
(101'Z) , (1Z21). (1100)]
l ·
and LE(A 1 ) -= {FZF)F 4 • F1F)F4 • F1FZF~l4' F1:Fz
4 Z
We consider a second (3 , :3 ) confounded design 0z whose generating
matrix is
0 1 1 1J
[
1 1 1
°
then L(A ) = {(0111). (1110). (Z001). (1Z21)}
Z
and LI(Az ) • {'2'3F4' F1F2F3, F1'4' '1F2F:l41 •
12
If we take
[U nJ
1
E =
o1
0 0
F
E~
then L£.(A ) = L (A2 ) and 01 is a design equivalent to D2 1
We now derive sufficient conditions for two matrices
to generate equivalent designs in the following 18llll'lla:
Lemma 4.1:
designs 01 and
°
2
Let .11 and A2 be two matrices generating the
, respectively.
A sufficient condition
for
01 and 02 to be equivalent is that we can wr1 te
Cit
A1 =
(1)
.12 On En
where ~ is a k x k non-singular matrix in GF(s),
On is a n x n diagonal matrix whose diagonal entries are
non zero elements of GF(s), and
En is a n x n pemutation matrix.
Proof:
L~.1l)
Given the hypotheses (1) we may write
= Ll(~
z
= t~
.12 Dn En)
I :g,-Ii L(Se A2
On En)}
z
=
{~ I ~6L(A2
because L(~ A2 DE>
1c
n n
matrix
Cit.
= L(.12
On En>}
DE) for any non-singular k x k
n
n
E bZ
Also because (En l> n- =
fe
Z
we can write
Z
LI(A1 )
={
Eb
(Enl) nZ
I
EJi.€ L(~ On)
= {(Enl)-b
I
h" L(A2 On> }
= { (Enl>b
I
0~1
but because (0- 1 ~)z
n
= ~z
z
lleL(A2 )}
we have that
J
13
which proves that 01 and 02 are equivalent.
5 - Modular representation 2! .:!:b.2 (sn, sk) confounded factorial designs
The modular representation was introduced by Slepian
C20J
in connection with problems in coding theory and independently by
Burton and Connor [5] for the case of factorial designs.
We shall de-
fine here this notion and give some results useful for the enUJlleration
of confounded designs that will be worked out in the next chapter.
k
n
We have just shown how a (s , s ) confounded factorial
design is related with its generating matrix A. where
11 a 12 ... a 1n
a
21 a 22 ••• a 2n
(5.1)
A
[.1.1 ~
...
a ij £ GF( s),
=.
• .•••
.t.J
=
~1 ~2 ••• ~n
We have shown that the (sk_1 )/(s_1) confounded parallel pencils
denoted P(QI) are the ones for which the vector
Q'
satisfies the
relation
Q' = ~'A
for some k-vector ~,
= () l'
~ 2'" I f
allow ~to be the null vector).
).
k)' ). if. GF( s) (we do not
Each such parallel pencil carries
(s-1) degrees of freedom.
Let us now think of the non-null column vectors
(i
=1,
!.:1
2, ••• , n) as the coordinates of points in PG(k-1, s).
FG(k-1, s) there are N
by P , P , , •• , P •
2
N
1
= (sk_1)/(s-1)
of A
In
different points that we denote
We denote the (kx1) null vector by PO.
Then with each generating matrix A we can associate a set denoted SA'
14
of n not necessarily distinct points taken fram
PO.Pt •••••PN •
We notice that the same set SA of n points may be associated with
many generating matrices which are characterized in the following
obvious lemma:
.2&1:
Lemma
Let At and 1 2 be two (lan) matrices
= SA if and only if we can write
2
~ Dn En
then SA
=
1
A1
where 0 is a diagonal matrix whose diagonal entries are non-null
n
elements of GF( s) and E is a pel'll'1utation matrix. Then all the
n
matrices which have the same set SA of n points generate equivalent
designs by Lemma 4.1.
Let us denote by
G~
the number of times the points Pi
A
appears in the set SA (i = 1.2 ..... N) and let GO be the number of
times the null point Po appears in SA.
••••
*' =
\ie define ~A
G~) as the modular vector of the design whose generating
matris is A.
Such vector determines completely the set SA and
thus all the matrices which have the same modular vector
generate equivalent designs.
~*
It is not true however that different
modular vectors correspond to different designs.
n
to ~
.
A A
(GO' G1'
We conclude that
= n.
design.
6
We give an example of the modular vector of a (2 • 2 3 )
Suppose that
1
=
·
1 0 0 0 1 OJ
0 1 001 0
[
001011
and let us denote the points of PG(2.2) as
follows
15
P1
Pc
· ~ m ~ [U ·
~U1. P2 ~ m· P3 ~m
~
P4
P5
Po
~ UJ ·
m;
then SA = { Pi' P2' PJ ' PO' P7' P
31
andEl' = (1,1,1,2,0,0,0,1) . i. the modular vector of the design.
Each k-vector
L'
in the equation (5.2) can be used to
define in PG(k-1, s) a (k-2)--flat given as the set {A I~' A =
hence the i
!'
!..i
=
th
coordinate b
i
of the
vector~'
\
= ~'
o}
A will be zero if
° that is if the point represented by !..i is in the flat
defined by~.
There are N
= (sk_ 1 )/(s_1)
different ~ denoted by
~1' ~2"'" ~N; to each ~ cooresponds a (k-2)--flat
Fi
Pi
={ X ,
= P(~,>
= °J and
l2.t = ~ ~A.
! ' ~
the confounded parallel pencil
where
We denote by Di -1 the order of the
interaction confounded by Pi(i
12' =
= 1,2, ••• ,N)
and define the vector
(01' 02' ... , ~)
as the vector of the orders of the confounded interactions.
We define the N x N incidence matrix B =
« Bij »
° if the jth point of PG(k-1,
1
lies on the i
A
.Jii = (G1'
GA , ... , GA ) whioh is the vector
2
N
with its first element left out.
Lemma.s..g:
be written as
th
otherwise
We define the vector -
gz'
s)
by the following:
The relation giving
~
in te1"llls of
g,
can
16
(5.4)
l:N
where
A
=n
Gi
A
- GO
G~ ~
i=1
(5.3) being over the reals.
Proof:
0
and the operations in
0i is the number of factors in the interaction
confounded by the parallel pencil Pi = P(~,> where ~
= ~ A or
h.t •
equivalently D is the number of non-null coefficients of
i
~
We know that the k th coefficient ~
!x of
is null if and only if
2t
the (k-2)--flat F contains the point of PG(k-1, s) represented by
i
!X.
N
but ~
A
B · G. is precisely the number of points of SA which
iJ J
do not lie in the (k-2)--fiat
N
~1
~ B
A
Fiand
= 1,2, ••• ,N
ij Gj = 0i,i
is equal to 0i.
We have
or in matrix form this can be written as
~
B !iA =]2. -
Let us premultiply (5.3) by Bt to obtain
N
Bt B !i = BI
A
.12.
Consider
Bt B
= «mij »
where
N
mii
=1:
k=1
I\:i I\:i
= the
number of (k-2)':'-flats not containing a fixed point Pi.
= (sk_ 1 )/(s_1) _ (sk-1_ 1 )/(s-1) = sk-1
N
mij
=1:
1\i I\:j
k=1
k 1
=!-=_ 2
s-l
( 5. 6)
Hence BIB
where I
N
= sk-2
of (k-2)--flats not containing
Pi or Pj •
k-1
k-2 1
k 2
-1) + s
- = s - (s-1).
s-1
s-1
IN + s k-2( s- 1) I N
is the N x N matrix of ones.
The equation
(
)
5.7
(s
= number
(5.,) becomes
k-2
s
! iA + sk-2( s-1 ) ( n-GA).
O ~
('oJ
= Bt
J2
17
where jN is a vector of ones.
Then
(5.8)
'"
~A
~
= k-2
-
A
(s-1) (n - GO) ~
which is the inverse formula to
Now given a vector
~
~
(5.3).
one would like to know if the
corresponding (sn, sk) design exists.
,..
If such a design exists then
we say that Q is admissible.
We have the theorem:
'lbeorem .5.a.1
tv
fi =
(G1' G , ... , G ) is admissible if and onlY if
2
N
N
R1
k'
2)
G is a non negative integer·
3)
1!l ~ >
Proof:
Gi
4: n,
1)
i
N
i = 1,2, ... ,N,
th
0 where ~ 1! the i
X.2!! 9I. B matrix
defined !D (5,21).
In a (sn, sk) design, the generating matrix has
rank k, which means that there exist k linearly independent columns
in the matrix, and the points of PG(k-1, s) associated with them are
said to be independent.
The existence of such a set of k independent
points is equivalent to condition 3), since
a (k-2)--flat
contains at most (k-1) linearly independent points we must have
.at '"~ ).
..
0 for i = 1,2, ... ,N and conversely if all the
a.r
"-
~
>
0,
then no (k-2)--flat contains the n points; thus there must exist
more than (k-1) independent points in the modular representation.
Conditions 1) and 2) assure the modular representation
associated wi th ~ to exist.
We say that a vector of orders of confounded interactions
~ is admissible if there exists a (.n, sk) design frClll which it can
18
be derived as above.
Theorem
12'
We have the theorem:
..2..1.
= (01' O2 , •• Of DN) is admissible if and only if
N D Ik-1
1) k sk-1!::
i - n s
i=1
2)
--L
k-2 [~~
~ _ §.:1
s
s
j']
N n
~
must b e a non-nega ti ye
= 1.
integer for i
3)
Proof:
N
'5'"
t:l
0
D
i
we have from 5.3 that
i =
D
i..
2, ••• ,N
j.N
~
B Q
A
= s k-1 I:N
i=1
A
G
i
k-1
= s
A
(n-G )
O
and oondition 1 of this theorem is equivalent to oondition 1 of
Theorem
5.1.
Using (5.8) we have that
B~ Q
=~
_ s-1 ., 0 = 1 [B' _ s-l j']
k-2
k-1 IN k:2 9. - _N
s
s
s
s
:Q
and oondition 2 of this theorem is equivalent to oondition 2 in
Theorem 5.1.
Obviously by (5.3) condition 3 of this theorem and
Theorem 5.1 are equivalent.
This proves the theorem.
6 - Necessarv oonditions for equivalent designs
Let us consider a (sn, sk) confounded factorial design
D1 with generating matrix Al and modular vector
Qi' = (G10 •
G11 , G12 • •••• G1n )
,N
= (G1o :Gi)
19
and a second design 02 with generating matrix A and modular vector
2
Q2' = (G20 •
''''
= (G201~2)·
£r..L.
Theorem
A necessary
designs 1§.
G21 • G22 • •••• G2n )
condition for 1)1
~
D
2
!d2. be equivalent
~
1)
= G20
G
,..10 ,.,
~1 =
2)
g,z
~ .!Q!: a pemutation matrix ~.
Unfortunately these conditions are not sufficient.
However. the
results of this theorem shall be useful in the next chapter for the
enumeration of confounded factorial designs by restricting the
search for different designs wi thin certain subsets of designs
satisfying the conditions 1) and 2).
Proof:
Let us now prove the theorem.
We recall that 01 and D , are equivalent if there
2
EF
exists a permutation matrix En such that I!(A ) = L n-(A ) where
1
2
Z
L~Ai) = {~ \
li£L(Ai )}
i = 1.2
th
th
th
We know that G10 columns. say. the i • i ••••• i
2
1
G10
the generating matrix A are null; then for every vector
1
we will have b
i1
= b i =. •• = b i
2
bZ
= O.
of
12£ L(A 1 )
In terms of the elements
G
10
F,
L,\A ) this means that the factors Fi • Fi • • •• , FL
will
1
1
2
15
10
z
never appear ina confounded interaction term ~ , all the other
F
Co
factors appearing at least once.
Similarly for the design D there will be G factors. say
20
2
• • •• Fj
b
G
20
F
which will never appear in Fe L£.( A ). all the
2
.
other factors appearing at least once in some tem.
Then if G -/: G it will be impossible to have
10
20
20
E F
L1:<A.1 ) = L n-"(A ) for any En matrix because it will be impossible
2
to map the G20 non-appearing factors of D2 into the G10 nonappearing factors of D1 •
Assuming now condition 1) to be true we prove the necessity
of condition
2).
~·"'en
the i th coo rdi nat e 0 f ~
"11
::L1 denoted G1i is non null
th
th
th
we know that there are Ga columns of A1 the i 1 ' i 2 ' ••• , i G '
1i
say, that represent the same point Pi roCk-1,s); i.e., any
two of these G column vectors differ from one another by a scalar
1i
multiple.
b
b
i1 i2
Clearly any vector ~c L(A1 ) will have its coefficients
b
to vanish simultaneously and moreover it is
G
1i
impossible to add any other coefficients to this set of G
1i
•••
i
coefficients without losing this property. In tem.s of the
_b z
F
interaction confounded F""" E. L (A.1 ) this means that the factors
Fi
1
Fi . . . Fi
2
G
always appear together in a confounded interaotion
1i
whenever they appear.
'!ben if it is impossible to find a permutation E for
n
which
~
= gz
I{ G1i I Ga
o . L_ k L_ n,
~
then we must have that
= k)1
(G
respectively).
1i
f.:
I{ G2j I G2j = k}1
for some integer k,
and G stand for the coefficients of !i and
2j
1
~,
In that case there is clearly no permutation matrix
F
~ on the fact~rs that will make L- (A ) = L
1
EJ
(A2 ) because the
number of sets of k factors that appears simultaneously is different
in both design for at least one k.
CHAPTER II
ENUMERATION OF (sn, sk) CONFOUNDED FACTORIAL DESIGNS
1 - Technique of enumeration
We have shown in the last chapter how the confounding
scheme of a (sn,sk) design is related to a (k x n) generating matrix
A:
the confounded pencils P(12') are the ones for which
12'
(1.1)
=l. i A,
each such pencil carrying (s-l) degrees of freedom belonging to the
interaction
b
Z
~.
We know also that each non-null column of a k x n
matrix represents a point of PG(k-l,s).
For a given design D with
generating matrix A one may associate a set of n points or less of
PG(k-l,s), called the generating graph.
The generating matrix
is not unique and the generating graph is also not unique for a given
design.
We consider the vector ~ in (1.1) as defining a (k-2)-f1at
in PG(k-l,s) defined bY{X
I ~,
X=
a}.
terms we can say that the pencil P(12) with
In these geometrical
12' defined in
(1.1)
will belong to the interaction F F ••• F
if the points
i
i
i
th th
1
2
r
represented by "J:.he i 1 ' i 2 ' ••• , i~h columns of A are the only points
of the generating graph not belonging to the f1at{! I .!.., ! =
o} .
For geometries of dimension 1, 2 or J, it is easy to plot
the generating graph of a design and to produce from it the confounding structure of the design.
This is best illustrated by the
22
following example:
We consider the design of class (s6, s3) whose generating
matrix is
°° Pi ° P2
° i ° q1 q2 °
[
°° 1 r 1 r 2 °
1
(1.2)
A=
This gives rise to the
~ 0, qi
0,
i
I:
I:
Pi
I:
O.
r
0.
generating graph:
.~
(1.3)
A point may represent one or more factors.
In the
ex~ple
above the point (1,0,0) represents two factors, 1 and 6, while the
other represent one factor each.
Consider now all the possible (s3_1 )/(s_1) lines (1-flat)
in PG(2,s).
The line containing the points associated with the factors
2 and 4 for example, correspond to a vector
which
!'
1.
I
= ( ~ 1 ,A 2' ~ 3) for
= ~, A has its coefficients b2 and b 4 equal to zero; the
corresponding pencil P(!2) belongs to the interaction F1F F F6•
3 5
The line containing the points associated wi. th the factors
2 and 3 contains 5 also.
Such a line corresponds to a ~ such that
£' =~, A has its coefficients b
= b = b = 0; the pencil P(£)
2
3
5
belongs then to the interaction F1F4F6• All the other lines are
analysed in a similar fashion to obtain the confounding structure for
the design.
This is shown in the following table of confounded
F
interactions (these being all the elements of LL{A»:
23
main
effect
first
order
(1.4)
second
order
third
order
fourth
order
F1F4F6 1
F2'3 F4'5(s-3)
'1'3F4'SFf, (S-2)
'2F4F5 1
F1F2F F6 1
3
F1'2'4F5F6 (S-2)
F2 F F4 1
3
F1F2'5F6 1
'1F2F3F5F6 (S-3)
F3'4F5 1
. F1 '3'5'6 1
F1'2F3F4'6 (S-2)
F2'3F5 1
fifth
order
'1F2 F3 F4 F5'6 (S2-.4S+5)
In table (1.4) the number at the right of an interaction
tem ·indicates how many pencils of that particular interaction are
confounded.
gene~al
In the
case of a (sn. sk) design with generating
k x n matrix we consider all the possible (k-2)-flats in PG(k-i.s)
and produce in a similar fashion the table of confounded interactions.
As we can notice. this method of enumerating the confounded inter..
actions is quite easy and visual for 1. 2 or 3 dimension.
For
higher dimensions such method loses its simplicity.
Let G be the generating graph of a (sn.sk) design with
generating matrix A.
Let QI
= (G 1 •
G •••• , G ) be the modular
2
n
representation of the design (as defined in 1.5). where Gi (i~ 0)
.
th
is the number of columns of A. which represent the i
point
PiE PG(k-l. s) • G
i
(i ~ 0) can also be interpreted as the number of
factors associated with the point Pi of the generating graph; GO
is the number of null columns of A.
Let us suppose that 0 Li/_
••• L i p (k , p ~ n)
are the only non-null coe1'fients of Q I . then the corresponding
generating graph G can be de.scribed as the set of points
24
1 w~ere the point Pi.J
{Pi' Pi ' ••• , Pi
1
and
t
J=1
2
P
Gi .
J
stands for G1
factors
j
= n-G O'
'",Te
now define two generating grauhs to be of the
structurl'l if there exists a mapping
t
~
of the points of one graph
onto the points of the other such that
(51)-
All the incidence relations of the graph are preserved by
the" mapping, or in other words if dim{X\
= dim{' (X)}
for any subset X of the points of the first graph.
The number of factors associated with Pi and
Theorem
!
Pi are equal.
L.1
~ (sn~sk) designs ~ equivalent if and only II their
graplEhave
~ ~
Proof:
structure.
1ve know that two designs with respective sets of
F
F
confounded interactions L-(A
) and L=-{
A ) are equivalent if and only
1
2
if one can write
F
EnE
L~A1) = L
(A 2 ).
But the set L!{ Ai) consi sts in the enumeration of the
incidence relations of the points of the graph with the (k-2)<flats of the geOmetry: (F
F
•• , Fir) E L~A1) means that t..""lere
ii i2
exists a (k-2)-flat which does not contain Pi' Pi ' ••• , Pi
or
12k
equivalently which contains
....P...
i-'~P"'i-'-'-'-'-'--Pi - (its complementary
12k
set in the generating graph). Hence the mapping of a generating
graph onto another, which preserves the incidence relations of the
graph is a necessary and sufficient condition for going from one
design to an equivalent one.
If we say that a class of equivalent designs forms a
~
of design, the last theorem states that there are as many types
25
of designs as
t~ere
are graphs with different structures.
One object
of this work is to enumerate the different types of (sn, sk) designs
for special values of s, n and k and their
confoun~ing
structure.
In order to enumerate all the different generating graphs
i.e., the ones with different structures we shall proceed in two
stages.
In a first stage we find the entire set of graphs or
point configurations such that for any pair of configuration in
the set no mapping exists such that S1 is satisfied.
In 2 dimension
we have to describe and enumerate all the different possible configurations of p points in the plane.
In a second stage we look at the number of ways we can
associate factors to each point of a given configuration such that
the generating graphs obtained these ways have different structure.
Such a procedure shall lead us to a complete enumeration of the
different types of designs.
2 - Construction of (s~,sk) confounded designs for the case k
Case k
= 2:
= 2.3.4
a(sn, s2) design corresponds to a choice of
n points in the geometry PG( 1, s).
correspond to the (s+1) O-flat.
The (s+1) pencils confounded
A confounded pencil belongs to
F F ••• Fir interaction if the points i 1 , i 2 , ••• , i r do not lie
i1 i2
in the corresponding O-flat. The 23 types of designs of class
(s6, s2) are enumerated in Appendix II.
Case k = 3:
n points in PG( 2 t s) •
a( sn, s3) design corresponds to a choice of
The s2 + s+ 1 pencils confounded correspond to
the (s2 + s + 1) lines in PG( 2, s).
Fi
r
A confounded pencil belongs to
interaction if the points i 1 , i 2 , ••• , i r do not lie
26
in the corresponding line.
ated in Appendix II.
Case k
They divide into 38 types.
= 4:
n points in PG(3,s).
The designs of class (s6, s3) are enumer-
a(sn, s4) design corresponds to a choice of
Here the (s4_1 )/(s_1) pencils confounded corre-
spond to the (s4_ 1)/(s_1) planes of PG(3,s).
A confounded pencil be-
longs to Fi . Fi . . . Fi interaction if the points i 1 , i 2 , ••• , i
r
1
2
r
6 4
do not lie in the corresponding plane. The 23 types of (s , s )
designs are enumerated in AppendixII.
In each class of designs, every type is described in the
.following synopsis.
The generating matrix is given in a canonical
form; the "x" r S in the matrix represent
a
non-null element in GF( s)
chosen such that all possibly non-vanishing sub-determinants of the
matrix must indeed be non-vanishing.
sub-determinants are indicated.
Special cases of vanishing
The generating graph is produced.
'!he various pencils confounded are grouped under the interaction
•
they belong to.
3 - Types of
tguiwent. d."gns
We will need the following lemmas:
Lema ;.1:
If.!. and
.!.t
= !2.Z
{:l ,
!2. are two n-vectors such that
then the sets
!2. r :I. =
01
tx
Z
I.!. r
!
= Oland
are identical.
The proof is by induction on the number of non-null elements of
!. and
12.
If.!. and
!2. have only one non-null element in the p th
•
27
I
position then t!le set{!Z
which equals {;[.Z
I ;[.'
o}
A' ! =
may be written as{!z,
X~
=
o}
I Y~ = o} .
:9. = o} = {;[.z
Let us suppose the lemma to be true when A and :9. are such
Z
z
that A = :9. '3,ch with (r-l) or fewer non-null coefficients. We now
investigate the case when exactly r coefficients of A and :9. are non
In that case
null.
{IZ I A' 1 • o} = {xZ
{;[.z I :9.' ;[. = oj = {;[.Z
(J.l )
(3.2)
But by
.(J.1)
{!Z
= 1,
j
Z
~ypothesis
ai Xi + ••• + ai Xi = o}
1 1
r r
bi Yi + ••• + bi Yi =
1 1
r r
of induction the following subset of
OJ.
I ai 1Xi 1 + •••
2, ••• r}
=0 ,
+ ai Xi = 0; at least one Xi.
r r
J
equals the following subset of (3.2)
I bi 1Yi 1+ •••
+ bi Yi = 0; at least one Yi. = 0, j = 1,2, ••• , ry;
r r
J
in particular both sets contain the same number of elements. 13ut
{i
the number of solutions of A ' ,2S; = 0 and
that the sets{!Z
A'! =
z
£' ;[. = 0 and Yi
t'X. I
j
a and
~
°
j
:9.'~
Xi.
~ a
= i,
2,
number of elements and hence are equal.
j
=
° being the same we have
= 1,2,
... , r}
•• .,
r}
and
contain the same
This terminates the proof.
Theorem 3.1
If
{.!:i.1
and {~1 are two sets of vectors with the same
dimension such that {~l
and
\:l
l:9.j;[.
= {Q~l
= 0 ill j}
then the sets!!Z
I Ai! = ° all i}
are equal.
The proof is similar to the one for Lemma 3.1.
Assume now a (sn, sK) design 0 with k x n generating matrix
A.
The (sn, sn-k) confounded design 0* with generating (n-k) x n
matrix A* such that rank A*
= n-k
and A A*'
= 0,
is said to be the
dual design of D. In other words D* is the design the confounded
pencils of which have their coefficients orthogonal to A.
28
Lemma 3.2:
The correspondence of a (sn, sk) design to its
(sn, sn-k) dual establishes a 1:1 correspondence between designs of
class (sn, sk) and of class (sn, sn-k).
Lemma 3.3:
signs are equivalent
Proof:
designs 0
for
The dual designs of two equivalent (sn, sk) dedesigns~
Let us assume that the two equivalent (sn, .k)
and. O have A1 and 1. as generatir.g :natrices and hence
2
2
1
F
BQIl1e
permutation matrix En we have
F
= L-(A
E ).
2 n
By Theorem 3.1 we have immediately that
F
.= L-«A
E )*)
2
n
(where * is defined as before).
2
(A E ). as equal to A E and
2 n
n
F
and
Dr is
E F
L.f.(A1 ) = L lr(A2 )
.
F
L.f.(A*1)
But we can take
F
L-(Ai) = L-(A2En )
2,
equivalent to O
These lemmas allow us to reduce enumeration work in
half; one just needs to study the case of designs of class (sn,sk)
k
J.
n/2 in order to find all the different types of designs.
Up to this point we have tried to set formal definitions
useful in the theory of confounded factorial designs.
We have defined
identical designs and have shown a method of enumerating the various
interactions confounded in a given design.
We would like now to
develop an algorithm to enumerate the various types of designs of
a given class.
We shall study this for the class of designs
( s n , s2) and 2n ,23) •
4 - The number of different tyPes of !Quivalent (sn, s2) design,
In Chapter I, we saw how the different (sn,
8
2
) designs
correspond to the different ways of choosing n points on PGC1, s) and
29
how to work out the confounded pencils in each case.
PG(l,s) may
be represented as (s+l) points (Xl' x2 ) on a line, two points
(Xl' x2 ) and (Y1' Y2) being identical when xl = ~ 'Yl' ~ = 1 X2
( ~ a non-null element in GF(s». As an example let the design with
generating matrix
1 0 x x x 0 X)
(
=A
0 1 x 0 0 x x
x being any non-null element in GF(s) such that all the possible
non-singular 2 x 2 matrices are non-singular.
Such a design can
·be represented by the graph in PG(1,s):
(1,4,5)
•
. (2,6)
where the i th. point 1s the one corresponding to the i th column of A.
We saw that the confounded pencils p(£) are the ones with coefficients
12' given by
\
12' = ( ~
Taking (}.1'
A.
l' ~ 2
) (1 0 x x x 0 x)
0 1 x 0 0 x X
2) as coefficients of a O-flat in PG(l,s) the
confounded pencil for this (1 1 , ~ 2) will belong to interaction
th th
. th
F. Fi ••• F. if the i 1 ' i 2 ' ••• , ~r
points do not lie in the
~1 2
~r
.
n
2
O-flat in question. We then observe that two (s, s ) designs will
be equivalent if the repartition of their factors is the same or
in other word s if in both generating graph the number of points
corresponding to k factors (k
= 1,2,
••• , n-l) is the same.
follows then that the number of different types of (sn,
It
i)
designs equals thenumber of ways one can arrange n objects in at
least two different cells (this is the case when none of the columns
of A are null) plus the number of ways one can arrange n-1 objects
30
in at least two different cells (this is the case when one oolu.n of
I. is null) plus the number of ways one can arrange n-2 object. in
at least two different cells and so on.
If we denote by Tn the mJIIlber of diUerent. type_ of
(sn, s2) designs and by V the nUlllber of ways one can arrange k
k
objects in at least t'WO different cell. 118 have that
n
(4.1)
. Tn =
Vk
k=2
then Tn
Tn- 1 + Vn •
L
=
The quantity (V +1) is the number of ways one can arrange
n
11
objects in one or more cells, two arrangements being identical
when the number of cells with k objects is the suae in both &I'l"&D«-Menta for k
=1,
The quantity V +1 is well CDlpated (12].
n
'!he number ot type. of (sn, s2) designs for 3 ~ n' 30
2, ••• , n.
is given in the table below.
The number of types ot (sn, s2) designs
n
Tn
3
4
5
6
7
8
9
10
11
12
13
14
15
16
3
7
13
23
31
58
87
128
183
2.59
359
493
668
898
n'
17
18
19
20
21
22
23
24
25
26
21
28
29
30
Th
1194
1518
2061
2693
3484
4485
5739
1313
9210
11105
14114
18431
22995
28598
.
n
3
5 - The !PUII!eration qf the tvpts of (s , 2 ) conf9UJ!led
dtf'm'
The generating matrix .l of a (2n , 23 ) design is a 3 ][ n
31
matrix with entries in GF(2).
The columns of the matrix A can be
considered as points in PG(2. 2).
The geometry PG(2. 2) consists
of seven points and seven lines:
a graphical representation of this
geometry is given in Figure 5.1.
The circle joining the points
=
(101). (110) and (011) corresponds to the line x1 + x2 + x
O.
3
We shall number the seven points of
PG(2.2) as shown in Figure 5.2.
too
We denote as before the modular
,
vector ~*A
= (GO.G1.G2.GJ.G4.G5.G6,at
corresponding to the generating
matrix A. where Gi is the number of
columns of A which represent the
....-------IL.------~olo
011
point Pi in PG(2.2)ji
= 1,
2, •••• 7
001
and where GO is thenumber of null
Figure 5.1
columns of the matrix A. We have
7
Gi ~ 0 and
G = n. Each column
i=O i
of A being associated with a factor.
2
we have that the point Pi is
associated with G factors which we
i
denote by G also. We obtain the
i
table of confounded interactions
by considering the seven lines of
PG(2.2).
The factors associated
Figure 5. 2
with the 4 points not on a line are
confounded in the interactions
corresponding to this line.
We get
the following table of confounded
32
interactions:
G2 + G + G6 + G
7
S
G + G6 + G + G1
7
3
G4 + G + G1 + G2
7
G + G1 +
S
G6 + G2 +
G +G +
(S.1)
7
3
G2 + G
3
G + G4
3
G4 + G
S
G1 + G4 + G + G6
S
where for example the term G2 + G + G6 + G corresponds to the
7
S
.confounded interaction with G2 + G + G6 + G factors. G of them
2
7
S
th th
th
being Fi Fi" • Fi ( say) where the ii • i 2 • •••• i G columns of
1 2
G
.
2
A represent the pofnt P2 = (100) in PG(2.2). Each term in(S.1)
represents the juxtaposition of 4 sets of factors (some sets might
be empty).
The addition sign is to remind ourselves that the G
i
inside an expression can commute 1. e•• G2 + G + G6 + G is
7
S
identical to G + G2 + G + G6 for example.
7
S
For example we have the (2 8 • 2 3 ) design 01 with generating
matrix
F F F F F F F F
1 2 3 4 S 6 7 8
1 0 0 1 1 0 1 1
1=
0 1 1
0 1
1
0 1
0 0 1 1 1 1 1 0
th
The Fi on top of the i
column of A helps us to remember that the
th
\
i
coefficient of the vector ~I = ~I A is associated with the
appearance of the factor F in the interaction to which the pencil
i
PCb) belongs. The modular vector of the design is
~rl = (0. 11. 12 • 03 • 24 • 1S • 26 , 17 )
33
·,There for
8xa."!l~)lo
2
':
refers to the 2 factors associated 'iTi t~ P4'
ta1Jle used to construct
t~1c
T::(,
confounded interaction is
t,
12 + 1S + 2 + 17
(
03 + 2~ + 1 7 + 1 1
I.}
1
2
2 + 17 + 1 + 1
T:1·) + 11 + 12 + 0 3
I.}
28 + 12 + 03 + 2
17 + 0.~. . + 24 + 15
1
1 + 24 + 15 +
I'
l
t~e
2
term 1 + 1 5 + 2° + 17 refers to the 5 factors interaction
n
Suppose we have two (2 , 2 3 ) confounded designs D1
F FI.} F F F 1
5 7 S
and D with modular vectors
2
5.1.1
5.1. 2
[~ven
by
Qi' = (go'
g1 '
"'*' = (h O'
h
··1 ' h 2 , h 3 , hi.}'
~2
respectively.
r:r
g2' '"3'
G4'
P'~
j
;' g6'
nS'
h
3"7)
6 , h7 )
,..
= (r.-
G' )
,~o
= ( hO
,w
lI' )
;'[e know from Theorem 6.1 of Chapter I tllat a necessary
condition for 01 anc J 2 to be identical is that
(1)
(2)
Cf
-'"
"0
- ··0
N
Q'
tV
= g'
E for some permutation matrix E .
7
7
These
conditions are not sufficient, however, and the problem of enumerating
t~e
various types of desi[';ns is now reduced to the following:
go and Q' we want to know the number of
of designs
J
7
~Tith
modular vector of the form G*
is a permutation matrix.
the different
t~rpes
Qvectors
given
of designs in the class
= ( go
:r tv
~
J ) where
7
By 'mumerating the various go and
(two ~ vectors being "different" if we cannot
obtain one from ti1e other by permuting its coefficients) we shall
n
obtain the total number of types of designs of the class (2 , 2)).
Theorem 5.1
The table of confounded interactions for D 1s given by
1
256
7
g2 + g5 + g6 + g7
3
6
7
1
g3 + g6 + g7 + g1
471
g4 + g7 + g1
5
1
2
g5 + g1 + g2
6
2
3
g6 + g2 + g3
7
3
4
g? + g3 + g4
145
g1 + e4 + g5
2
+ g2
3
+ g3
4
+ g4
5
+ g5
6
+ g6
!nd the table of confoynded interaction for 02 is given by
5.1.5
7
7
£" gi =Z
i=1
~'=
hi
and
1=1
H'En
for a pemutation matrix
E11
and a necessarY' and
Let us first illustrate this by an example.
second (2
8
f
Consider a
23) design D with its modular vector equal to
2
')
22' = (0,
then
t~e
1
2
2 , 1 ,
O~,
4
6
2 , 15, 1 , 17 );
table of confounded interaction is
6
2
1 + 15 + 1 + 17
6
1
03 + 1 + 1 7 + 2
4
1
2 + 17 + 2 + 12
15 + 2
1
6
1 + 12
2
+ 1 + 03
4
+ 03 + 2
+ 24 + 15
17 + 03
6
4
1
2 + 2 + 15 + 1
'and it is obvious that no permutation of the superscript will make
this table identical to ~~e one for D1• For example the first line
6
2
1 + 15 + 1 + 17 has no counterpart in the first table.
Let us prove the theorem.
The necessary and sufficient
condition simply says that by changing the names of the points in
the generating graph of one design we can make tables 5.1.4 and 5.1.5
identical.
This signifies that we can establish a one to one
correspondence between the 7 sets of gi factors in D and the 7 sets
1
of h j factors of D2 which will make tables 5.1.4 and 5.1.5 identical
and hence will establish the equivalence of D and D •
1
2
When it is possible to make two tables identical by
permuting the superscripts of the entries of one of the tables we
shall say (loosely speaking) that the two tables are similar.
We now proceed to enumerate the various disjoint subsets
of deSigns with
~'
=(G1 ,
G2 , G , G4 , G , G6 , G ). For each subset
7
3
5
we will characterize the different sets of similar designs and we
will give an algorithm to count the various types of designs.
Note - We shall denote by T the number of types of
n
36
n
(2 , 23 ) designs and by TO the number of types of (2 n , 23 ) designs
n
which GO value equal to zero.
Then we have the relation
o
Tn =Tn +Tn- l' T =1
3
n
Tn =
T~
1:..
i=3
In the following we shall only derive the value T~ and using
(5.1.6), T will be oomputed easily.
n
Step 1 - Three Gi's are non null
\-le assume here that the generating 3 x n matrix A with
·its entries in GF(2) has rank three and is such that none of its
columns is null.
We assume also that in the modular vector
f'J
Q. defined in (5.2) exactl.Y' three coefficients Gi are non null. This
means that the generating graph will consist of three non-oollinear
vertices or points one of which will stand for g1 factors another
'3
for g2 factors and the third one for g'3 factors and ~ gi = n.
We easily rea.lize that there is only one configuration of three noncollinear points possible, this meaning that different choices for
the three points of the graph will lead to equivalent designs.
Let us choose then the three non-collinear points to be
P1,P2)P3 and G1 = g1' G2
= g2'
'3
G
3
= g3'
say; then we have Q.'
L
gi = n.
i=1
The generating graph is reproduced in Figure 5.3 and the
= (g1,g2,g3'o.o.O.O),
table of confounded interactions in (5.3)
37
G2
G, + Gl
Gl + G2
Gl + G + G
2
3
G2 + G
3
1.M:':._ _--:~::::::
G,
.::i4
it '-ac1:o("S
Gl
Figure .5.3
where for example Gl + G2 means that one confounded interaction
.is composed of the G1 = gl factors placed in point 1 and of the
G2 = e2 factors placed in point 2 of the generating graph. From
Table 5.3 it is obvious now that any permutations of gl,g2,g3
(as coefficient in the vector G) will leave the set of values in
5.3 unchanged; for example. if we had considered the vector Q'
= (g2,gl,g3'O,O.o.O) instead we would have obtained the same set
(.5.3) corresponding to the set of confounded interactions.
We must then conclude that the designs under consideration
3
here are characterized by the sets gl' g2' g3 where~ gi = n,
i=1
n
gi~
i = 1,2,3. The number 51 of different types of designs
°
(with three Gi's non null and GO=O) considered here equals the number
of partitions of n into 3 parts.
Let us denote by p(n,k) the number of unordered partitions
of n into k
p~rts
i.e. p(n,k) is the number of solutions in
posi tive integers Xi of
n = Xl + x 2 = ••• +
then
(5.4)
5~
= p(n,3).
Jlk
for
Jlk ~
x k_l ~
• ••
~
Xl
~
1;
38
~
Gi ' s .!t! non null
We have to distinguish two cases corr.sponding to two
Step 2 -
configurations of the generating graph.
In the first case A has rank three, its corresponding G
vectors as defined in
(S.2) has exactly four Gi's non null and its
generating' graph consists of four vertices (out of the seven
possible) no three of them collinear.
We know that one needs to
study only one realization of such graph, any other with the same
configuration leading to an equivalent design.
Let us then
-consider the vertices 1,2,3 and S of the graph and let us also
assume that the n factors are divided into four sets with size
,
g1,g2,g3,g4 respectively (we have
vector is
4
~1
g1= n" gi> 0).
Then the
Q
,..
Q'
= (g1' g2' g3' O. g4' 0, 0)
i. e. G1 = g1' G2 = g2' G = g3' G = g4' The generating graph is
3
S
illustrated in Figure S.4; the table of the confounded interactions is
given by
G
2
+G
S
G + G1
3
G + G
1
2
G + G1 + G2 + G
S
3
G2 + G
3
G +G
3
G1
1 "".~_....:::::;t:::::::.. _ _::::ttt
S
8i
+G
S
Figure S.4
Looking at the Table 5. S we notice that an,y ptlJ"DlutaUon.
IV
of g1,g2,g),g4 al ooertlcients of ~ will leave (5.5) unohanged.
,9
Hence
t~e
''---
desifn
consider~d
here is
characterize~
by tr.A giving of the
L
f
= n; ei :> 0.
i=1 i
The number 3
of the different types of such
12
f1 ,17,2' I! '3 , f!l.j.
set
where
nesi~ns
is
S~2 = p(n,4).
(5.6)
In the second case, the generating matrix A has rank three,
the
-
f'I
- G vector
correspondin~
has four of its coefficients non null but
now the generating graph consists of four vertices three of which
are collinear.
Without loss of generality we can study the case when
these four points are P1,P2,P"P4 and take the
ftJ
Q'
= (g1,g2,g"g4'O,O,O)
i.e. Gi
tv
Q vector as
= gi
the generating graph is illustrated in Figure
5.5
i
= 1,2,3,4;
and the table of
the confounded interactions is given as
,~
gL
Fi'gure 5.5
(5.7)
G + G1
3
G4 + G1 + G
2
G + G + G
2
1
J
+ G + G4
2
.3
G + G4
3
G
G1
+ G4
'!he question is now that given the four integers g1,g2,gJ and g4
40
the sum of which equals n, what the pemutations of these number in G
are ,that leave the set of the interactions confounded given in
(5.7) invariant.
Using the graph and the equation (5.7) we see that
pemuting g1 g3 and g4 together does nct change (5.7) while on the
other hand it is impossible to interchange g2 with only other
coefficient.
~ving
Such kind of designs are thus characterized by the
of the coefficient g2 and the set
g1,g3,g4
subject to
~ gi = n. If we denote by S~2 the number of different designs
i=1
considered here we have
n-3
n-1
(5. 8)
S~2 =
p(n-m, 3) =
p(m,3) for n ~ 4;
1:
'2.
m=1
m=3
these two cases are
in G.
th~
only ones possible when four G are non null
i
'Ibis can be seen by enumerating the 35 possible sets of four
points.
step '3 -
fin
Gi are non'null
In this case the G vector has exactly five of its
coefficients G different of zero; the generating graph consists of
i
five points.
There is essentially one structure for such a graph
which can be described as two intersecting lines of three points.
can suppose the five points of the graph to be P1,P2,P3,P4 and P
5
f"I
and the Q vector to be
/'oJ
Q' = (g1,g2,g3,g4,g5'O,O) ; Gi = gi i = 1,2, ••• ,5.
The generating graph is illustrated in Figure 5.6 and the table of
the confounded interactions is given in (5.9.
Figure 5.6
We
41
G2 + G
5
G) + G1
S+
G
G
2
G4
+ G2 + G)
+ G + G4
J
J +
G +
1
G
G1
G4
+G
5
G4
+
G
5
+ G1 + G2 •
The problem is now to detM"lTline what kind of permutations we can
make on the gi's for the set of values
"is apparent from the Figure
5.6
(5.9) to remain unchanged.
It
that we can
- interchange
- interchange
gl and gj
- interchange the sets {g2' g
51
and \ g1 •g)l
and these are the only permissible moves allowed on the
in order to preserve the type of the design.
gi's
Counting the permutation
not in the above permissible set will give us the number s~ of
different types of designs of this kind.
n-4
(5.10)
where
S~
[xl
=
L
m4=1
1(n~)/2J
L
p(m.2) p(n-m4-m.2) .. n
m=2
~5
is the greatest integer less than or equal to
x.
~
Step 4 - Six G ' s .!!:!. non null in Q
i
In this case the matrix A has rank three as before but
rJ
the corresponding Q. has exactly six of its coefficients G non null.
i
The generating graph consists of 6 points.
It is easily shown that
essentially one kind of structure exists for a graph of six points.
It 1s characterized by the following:
- there are four lines containing three points each
42
- every two lines meet in one point.
For example one can consider the six points Pl ' P2' Pj' P4' PS. and P6
and
..Q, =
(g1.g2.gj.g4.gS.g6'O) i.e. Gi
= gi
i
= 1.2 ••••• 6.
The
graph is illustrated in Figure 5.7.
Figure S.7
The confounded pencils are found from the following table
(S.11)
G2 + G + G6
S
Gj + G6 + G1
G4 + G1 + G
2
G + G + G + Gj
1
2
5
G6 + G2 + Gj + G4
Gj + G4 + G
S
G1 + G4 + G + G6
S
the set of permutations of the
~tS
that leave unchanged the set of
values (S.11 >. is large and it is not obVious as in the previous cases
n
how to use it to obtain the number S4 of different types of designs.
We shall div.1de the set of such designs into disjoint subsets and do
the enumeration of types of the designs for each of them.
Let us
consider first the set of designs for which the associated vector
{;it
= (g1.g2.gj!g4.g5.g6'O)
that is one can write
is such that g1 ,,~
i
= 2.j,4.5.6;
43
tV
Q'
where d
i
.),.
= (g1,g1,g1,g1,g1,e1'O)
° for i
= 2,3,4,5,6
and
+ (O,d2,d3,d4,d5,d6'O)
6
I:
i=2
d = n-6g 1 •
i
This means that
we can construct the generating graph (equivalently the design) by
as signing g1 fa ctors to t.l-J.e point P1 ' g1 other factors to P2 and so
on for the six points; this takes care of 6g
On top of that we assign d
i
= 2,3,4,S,6
i
1
factors of the design.
additional factors to the point Pi for
to obtain the complete graph.
We can better illustrate what this means in tems of
confounded interactions calculated from t.l-J.e Table 5.11 if we introduce
the notation
Gi = G1 + Di
i
= 2,3,4,S,6
and rewrite the Table 5.11 as
(5.12)
3G 1 + D2 + D + D6
S
3G 1 + D + D6
3
3G1 + 04 + D2
4G1 + D + D2 + D
S
3
4G1 + 06 + 02 + 0'3 + 04
3G1 +
0)
+ 04 + DS
lJG 1 + 04 + DS + 06.
In this table above we have to interprete the first term
3G + D + D + D6 as representing an interaction of 3g + d 2 + d +
2
1
1
S
S
factors usually written as G2 + G + G6 • The second tem 3G 1 + 03 +
S
represents an int~raction of 3g + d + d 6 factors usually written as
3
G1 + G + G6 and so on for the other tems in (.5.12). If we fix
3
G1 = gl J.. gi and look for the permutations of g2,g3,g4,gS,g6,wh.ich
will leave (5.12) unchanged then it is equivalent to look at the
d6
D6
44
permutations of d2,d3,d4,d5,d6.which leave (5.12) unchanged.
If we do not consider the G1 's in (5.12), the set of terms in (5.12)
is Similar to the one in (5.9); this is because the points
P2' P3' p4' P5' p6 which contain the term di have the same structure
as the ones considered in step 3.
But the set of permuu.tions
of drs leaving (5.12) unohanged must also leave (5.12) where the G1 's
do not appear. unchanged.
Hence we can just consider the permutations
seen in step 3 and select the ones that leave (5.12) invariant.
There are two independent permutations namely,
. (5.14)
1 - transpose 3 and 6 and transpose 2 and 4
2 - transpose 2 and 6 and transpose 3 and 4.
faldng that into account we can anumerate the various types of
(2 n , 23 ) designs of the kind defined above.. We have to notice at this
point that the fact that g1 is the smallest
restriction:
~
value is not a
we would have obtained a simi1&r set of equations had
we assUMed ~ other coordinate to be the smallest.
Let s~ be
n
the number of types of (2 ,2 3 ) designs urder aonsideration.
We have at first that
{n/61
n
s1 =
n-6g -4
1
2-
L.
g1=1
d5=1
0
f(n-d -6g )
1
5
for n
~
11
otherwise
where f(x) is the number of ways to place x factors in the four
points P2,P3,P4,P6 when one does not di stinguish two arrangllllents
obtained fran one another using the permutations described in 5.14.
'!he formula (5.15) can be explained as follows:
- we first distribute g1 factors on all of the six points
to P6 thi s taking
ca~e of 6g1 4 n factors. Hence 1 '- g1 '
( n/6
Ii
I .
45
- we then add d
factors to P5' this point not being subject
5
to any permutations with the others. We still have to ~istribute the
remaining n-d -6g factors on the four points P ,P ,P ,P and this
1
2 3 4 6
5
is done in f(n-d -6g ) ways.
1
5
The formula for f(x) is
[x/2]
(5.16)
f(x)
=
~ q(R.2) [2q(x-R,2) + r(s-l,2)]
1=2
[x/21
L.
+
r( 1,2) p(x- i,2)
t=2
-where q(n,m) is the number of partitions of n into m unequal parts
r(n,m) is the number of partitions of n into m equal parts
p(n.m) is the number of partitions of n into m parts.
'!he formula is explained as follows:
- for each value.1 , split it into 2 unequal parts that
are assigned to (P ,P4 ) or to (P ,P ); this can be done in
6 2
3
q(~ ,2) ways.
- the (x-i) other factors are divided into two unequal
or equal parts which gives the multiplicator
- if
J
f
2q(x-j ,2) + r(x... ,2) •
is split into 2 -equal parts then the order of the
two ot..l-J.er parts coming from the other x-l fa.otors does not matter
and we count r(j,2)
p(x-j,2) different partition in that case.
All these formulae q(n,m),
p(n,m) are tabulated in the volumne
"tables of partitions" published by the Royal Society (1958)
[12].
The next class of designs we want to study is the one with
N
a Q vectors with six coefficients non null two of which G and G
i1
i2
(say) are such that G
G L G
i f: i 1 or i ; that is two
i
i
2
i
1
=
2
of the non null coefficients are equal and of smaller value than the
others.
Obviously this olass of designs is disjointed froJll the
previous one we have just studies.
Let us suppose here that G1 = G LG
i
2
i = 3.4,s.6.
Then
one oan wr1 te
6
1-
d = n+6g ; d ~ O. As previously we use g1 + d to denote
1
i
i
i
i=3
th
the gifaotors associated with the i
point. We have the table
with
of interaotions wr1 tten as
3G1 + DS + 06
3G1 + 03 + 06
3G1 + 04
~1 + Os + D3
4G1 + 06 + 03 + 04
3G1 + 03 + 04 + Os
4G1 + 04 + Os + 06
where again we have pooled toegther the various G •
1
The term
3G + Os + 06 is not as well defined as before beoause it 1U.y
1
oorrespond to G + G + G or to G + G + G6 , but if we decide to
t
6
2
5
S
N
fix G1 and G in Si. with the property that G1
2
=G2
L
Gi
i
= 3,4,s,6
and look at the permutations of g3,g4,g5,g6 that leave invariant
,
(5.11) it is sufficient to look at the permutations of d3,d4,d~,d6
that le.ves (5.1?) invariant.
Here again if we delete the G1
faotors in (5.1?), the table beoomes identioal to (5.?).
The argulllent
goes exaotly as befOre to obtain the set of permuUtions by looking
at the generating graph given in Figure S.8 below.
47
Figure 5.8
The only permissible permutation is to transpose 3 and 5.
,The number of types of such design s2n is
for n
(5.18)
~
10;
this is seen as follows:
- we put g1 factors on each of the 6 points P1 to P6 •
- there is n,;..6g 1 factors remaining,
- we choose f of them to be placed on P and P • Because
5
3
we can transpose P and P there are p(f,2) ways of having the f
3
5
factors placed,
- we finally split the remaining (n-6g 1-f) factors in two
orderly groups to be placed at P6 and P4 •
Let us notice here that if we had started for example with
....
Q. vector with G1 = G4 L Gi i # 1,4 we would have obtained a similar
class of designs. this can be seen by producing the table of confounded interactions or by noticing that the generating graph of
such a design is of the same structure as the one in Figure 5.8.
N
But if we take Q. with G1 = G we obtain a different design.
5
We have to study this class of designs where G1
i = 2,3,4,6.
~
= G5
L G
i
We can write as before
= (g1,g1,g1,g1,g1,g1'O) = (0,
d2 ,d ,d 4 ,O,d 6 ,O)
3
and the table of interactions is
3G1
+ 06 + 02
3G1 + D3 + 06
3G 1 + 02 + D4
(5.18)
4G1 + D2 + D
3
4G1 + D2 + D3 + D4 + D6
3G1 + ~3 + D4
°
4G1 + 4 ,+ D6
and it is obvious that this table is different frcn
(5.17). We
find here that three independent permutations of the d
i
I
S
leave
(5.18) unchanged:
1 - transpose 2 and 3
2 - transpose 6 and 4
3 - transpose 3 and 6 and transpose 2 and 4;
then
\(n-6g 1 )/2)
~
P(f,2) p(n-6g -f,2) for n odd,
1
~
n ~ 10
1
l(n-6g 1 )/2- 1
1n-6~4)
p(f,2) p(n-6g 1-f,2)+
f
L
1:.
f=2
f=l
for n even,
n
~
10.
We now have to study the case when three of the
N
coefficients of
~
equal the smallest G
i
I
8
value.
The argument i.
similar to
t~e
one given previously and we shall only summarize the
steps.
There are essentially three situations to consider.
The
N
first one consists of the designs with Q. vector of the fom
[, = (g1,g1,g1,g1,g1,g1'0)
+ (0,0,0,d 4 ,d ,d 6 ,O)
5
with a generating graph illustrated in Figure 5.9 below. The table
of confounded interactions is given below in (5.20).
1.
,g,
.Figure 5.9
3G1 + D + D6
5
3G 1 + 06
3G1 + 04
%1 + D
5
(5.20)
40 1 + 04 + 06
3G1 + 04 + 05
40 1 + 04 + 05 + D6
the only pemutation possible which leaves (5.20) '.lnchanged is the
transposition of 4 and 6.
designs is
(5.12)
The number of different types of these
50
N
The second case is the class of designs with the Q vectors equal to
N
Q' = (g1,gi,g1,g1,g1,g1'O) + (O,O,d 3 ,O,d5 ,d6 ,0)
and the table of confounded interactions is
3G1 + D5 + D6
3G1 + 06 + D3
3G1
ltG 1 + U + 03
5
4G1 + 03 + 06
3G1 + D3 + D5
ltG 1 + D5 + 06
•
All permutations of 3, 5 and 6
leave the above table invariant
In/6)
(5. 22) s~ =
p(n-6g1'3)
L
g1=1
f or n .
'"
a
1
i\
8s: I,.. ~"
Figure 5.10
The third case is the one in which
N
Q = (g1,g1,g1,g1,g1,g1'0) + (d 1 ,d 2 ,0.d 4 ,0,0,0);
the table of confounded interaction is
9•
51
3G1 + 02
3G 1 + 01
3G t + 01 + 02 + 04
4G1 + 01 + 02
4G1
+
D2
+ 04
3G1 + 04
4G1
+ 01 + 04
and all the pel"ll1utation of g1,g2 and g4 as coefficient of
IV
~
will
leave the above table invariant:
[n/6]
(5.23)
s6 =
la
=
p(n-6g1'3)
55
g1=1
We now consider the case when the vector
to equal the smallest G value.
i
IV
~
has four of its coefficients
We shall distinguish two cases.
First we can take the four equal values as G1 + G2 + G + G4 = g1
3
and we get the table of confounded interactions as
3G1 + D5 + 06
3G 1 + 06
3G1
4G1
+ 05
4G 1
+ 06
3G 1 + 05
4G1
[n/6)
L
p(n-6g1'2)
+ 05 + D6
for n ~ 8;
g=1
in the seoord case let G1 = G4 = G • G6 = g1; then the table of
S
confounded interactions is
52
JG 1 +
G
2
3G1 + G3
3G 1 + G2
4G1 + G2 + G
J
4G1 + G2 + G
J
JG 1 + G
J
~.
4G1
In/6)
5.25
s~
= 1.
p(n-6g ,2) :.
1
S7
n ~ 8.
g1=1
We have now to consider
the case when five non-null coefficients
AI
of
~
G
1
=G2
are equal and smaller than the last one.
= G = G4 = G, = g1.
J
Let these to be
The table of confounded interaotions is
3G1 + D6
3G1 + D6
3G
1
4G1
4G1 + D6
3G1
4G 1 + 06
for n
(n/6) -1
o
~
7
otherwise.
Finally the case of all G equal, produces one more type and
i
= n/6
5.261
s~o = = 1 if In/6
= 0 otherwise
.
n
The total number S4 of types of designs studies in step 4 is the
J
n
n
~~um of s1 to s10 i.e.
10
Sn4 --
1:
i=1
•
53
= 1,
The values of s~
i
given in Table 1.
The values of 54 are also obtained.
••• , 10 were calculated for n& 30 and are
n
Step 5 - The seven gi (i ~ 0) are non WI
In this final case we study the designs whose generating
fIJ
matrix A is of rank 3 and whose associated Q vector has all of its
coefficients different from zero; that is
wi th G ~
i
Q = (G1,G2,G3,G4,G5,G6,G?)
?
0 i = 1.2•••• ,? and '2: G = n.
i
The generating graph of
i=1
such design is illustrated in Figure 5.11.
The table representing the
confounded interactions is given below:
G2 + G + G6
5
G + G6 + G?
3
G4 + G? + G1
G + G1 + G2
''---
5
+ G?
+ G1
+ G2
+G
3
G6 + G2 + G + G4
3
G? + G3 + G4 + G5
G1 + G4 + G + G6
5
Figure 5.11
The set of permutations that leaves the above table unchanged is
rather large and it is not obvious how to use it in order to obtain
a formula for 5~. the number of types of the designs described above.
We shall use a similar procedure as before:
separate the set of
desiglJ,s into subsets and perform the enumftration wi thin each subset.
We shall g1ve the full detailed argument for the first few formulae
and give only the main steps for the others.
IV
We first consider the case when one coefficient of Q , say
0i' is such that 0i I.
°
the minimum 0i value.
Let 8Uppo.e that 0t i. the wi. . . ooett1a1ent.
j
ItIl
j~ 1, i. e. one ooetfto1ent of i .tt.a1u
then on. can Wl'1te us1ng notatian 1fttroc:luaed betON that
~ • (11,11,11,11'11'11,11) + (O,d2 ,d),d4 ,d"d6 ,d1)
where
di~
0
1
~
«\.
"'111.
!he .et of ocmtCNftded inteNet1Oft.
i. now written a.
401 + ~ + 1)S + 1)6 + ~
1fG1 + D) + D6 + ~
U;
4G1 + D,. +
+~
401 + 1), + ~ + D:3
4G1 + D6 + ~ + ~ + D4
JM]1
+~ +
n, + D4 + D.5
4G1 + D4 + DS + D6
•
We notic. that except for the t.1'Il 401 the Table .5.2.5 ilid.ntica1
to tab1•
.5.11.
H.no. it we t1x 01 • Cland 10«* tor the ,peftlUtat1onl
ot g2 up to 17 that leave the tab1. of oanf'OUJded int.raotion.
U1')l,l';:',J'
changed then this is the .... al alking for the I.t of pemutatiou
of' d 2 up to d leaving (.5.2.5) unohanced whioh in turn il the I . . al
7
asld.ng for the ..t of p8Jl111UtatiOM 1ea'f'1ng fab1• .5.11 unohepd. '!hen
it we denote t~ the mabel' ot typel of d..s.cn. de.ortbecl h.N ".
haye iDed1ate1y that
.5.272
wh.r.
tor n
54 il the mllber of t.Jpe' ot clll1pa b
i
lltep r.
f$M11I'
,
,
v. have to notio. that the taot tha' the IIlId..l 01 ~ . . ~,~
al 01 11 not a r ••tI'1ot1C1ftI &IV' . . . . . . . . WftJd ....
.
.
'
.
W
te.
Mt,
I
'
~
•
.•
ofequat11:tal dJI1lar to th• • • 1. ~,.2') *loh . . . . ~t 1t WI
,
!
•
55
sufficient to study only the case above.
n
The list of the t 1 values is give for n
~
30 in Table 2.
The second case we have to study is whenever two coeffi#oj
cients of Q equal the minimum value of the coefficients G •
i
If
we let these two values to be G and G we have
6
7
tV
Q
where the di
= (g6,g6,g6,g6,g6,g6tg6)
~
0
i = 1, •• .,5 and
+ (d1,d2,d3,d4,d5'0,0)
5
-
L di
= n-7g 6
1
The table of confounded interaction is given by
4G6 + 02 + 05
llG 6 + 0) + 01
4G6 +D4 + D1 + D2
4G6 + D5 + 01 + 02 + 03
4G6 + 02 + 0) + 04
4G6 + 0) + 04 + 05
4G6 + 01 + 04 + 05 •
We notice that Table 5.27 is identical to Table 5.9 if we ignore the
40 6 terms.
Th.en the permutations of d
that leave invariant
5
(5.27) will leave invariant (5.9) and conversely, hence t~ the
1
up to d
number of types of desings of this kind is g1ven by
5. 28
t~ =
In/7)
L
g=1
sn-7g
3
where s~ is given in (5.10).
for n ~ 12
The values of t
n
2
n')O are, given in
Table 2.
N
The case of three coefficients of the vector Q equal to
the minimal G value is easily treated.
1
There are two cases
corresponding to the two cases we meet in step 2 and the formula for
in/7)
.5.29
I.
for n
~
11.
g=l
The values of t~ are given in Table 2.
For the case of four coefficients ot the vector
fU
~
equal to the minimal
G value we have to distinguish two cases.
i
The first case is the one of G4
i
= 1,2,J.
= G.5 = G6 = G7 L
Gi
Then the table of confounded interactions is
4G7 + 02
4G7 + OJ + 01
4G 7 + 01 + 02
.5.30
4G7 + 01 + 02 + 03
4G7 + D2 + 03
4G 7 + 03
4G7 + 01
If one ignores the
4G7
•
in each term the last table is
identical to (.5.J) and we have
In/7)
.5.31
L
In/7}
L
s~-7g =
n
~
10
gal
g=l
the second cas. is when G1
p(n-7g,3)
= G2
+G
3
+ G.5
of confounded interactions is
4G1 +06 + ~
4G1 + 06 + D7
4G1 + D4 + D
7
lfG1
4G1 + 06 + 04
4G1 + ~ + 04
4G 1 + 04 + D6
G
i
i = 4,6,7.
The table
57
and
[n/7]
L
5.32
p(n-7g,3) =
g=l
t4
for n
~
to
all these values are computed in Table 2.
When now five coefficients are equal and smaller than the
other two ooefficients we have one case which is illustrated by
G
t
=G2 = G3 = G4=G5 ;
the table of confounded interactions is
4G7 = 06 + 07
4G 7 + 07
4G
7
1lG
7
4G7 + 06
4G7 + 07
4G + D6
7
In/7]
5.33
t
6= Lg=1
p(n-7g,2)
When 6 or more coefficients are equal
5.331 t~ = [n/7]
n ~ 7.
The total number s~ of types studies in this stage is
7
L
n
tj
•
j=1
We now summarize all the formulae in the following
"
.58
Summary of the formulae
5.16
n
= p(n,3)
5.4
Sl
5.6
S~2
5.8
5.10
= p (n,4)
n-1
S~2
m=3
n-4
n
51
m ;!
=
n > 4
[[n-m )/2]]
4 Y p(m,2) p(n-m -m,2)
4
m=~
l
S~ =
n > 4
p(~,3)
= l
4
5.15
n > 3
[[n~6]]
n- 6al- 4
gl=l
ds =l
2
L
f(n-d s-6g 1)
n > 5
n > 12
[[x/2]]
5.16
l
f(x) =
q(l,2) [2q(x-l.2) + r(x-l.2)]
l=2
[[x/2]]
+
l
r(l,2) p(x-l.2)
l=2
n
5.18
5.19
s2 =
n
53 =
=
[[n/6]] n-6g 1-2
I
gl=l
l
p(f.2) (n-6g 1-f-1)
f=2
[[n/6]]
[[(n- 6al)/2]]
g=1
f=2
I
2
n
p(f.2) p(n-6g 1-f.2)
[[n/6]]{ [[(n-6il)/2]]
I
2
g=1
p(f.2) p(n-6i 1-f.2)
f=2
f
f}
+ [[(n- 6 1)/4J]
f=1
even n > 10
>
11
odd
59
5.21
n
s4 =
5.23
n
s5 =
5.24
5.26
[[n/6]]
I
(n-6~ -2)
1
d5=1
gl=1
[[n/6]]
I
gl=1
n
s n7 -- s 8 --
p(n-6g -d 5 ,2)
1
n
p(n-6g 1 ,3) = s6
L
g=1
p(n-6g 1 ,2)
n
n
s9 + slO = [[n/6]]
n
>
6
into rn unequal parts
r(n,rn) = no. of partition of n
into rn
p(n,rn) = no, of partition of n
into rn parts
5.271
5.272
equal parts
greatest integer less then or equal to
54n
n
10
I
= i=l
s.n
x
n > 6
1.
=
n >
13
5.28
n
12
5.29
n > 11
5.32
n > 10
t.1.
>
[[n/7]]
5.33
I
g=l
> 9
n
>
8
n
>
7
-
[[n/6])
q(n,rn) = no. of partition of n
[[x]]
n
p(n-7g,2)
n
>
9
60
n
5.331
t 7 -- [[n/7]]
5.34
5n5 --
7
I
j=l
n
t~
J
TO = 51n + 5n + 5n + 53n + 54n + 55n
n
12
22
>
7
61
n
Table 1 - Number of (2 , 2
'.l
J
)
Designs
n
3
1
4
1
1
1
5
2
1
2
1
6
3
2
4
2
7
4
3
7
5
8
5
5
11
9
9
7
6
16
18
10
8
9
23
28
11
10
12
12
11
15
31
41
45
65
13
14
18
53
14
16
23
15
19
16
1
1
1
2
1
3
1
4
2
7
3
6
97
6
13
4
9
67
132
11
22
8
12
27
83
182
23
34
10
17
21
34
102
238
37
51
18
22
17
24
39
123
316
64
73
21
29
18
27
47
147
400
95
102
32
36
19
30
54
174
510
148
138
39
45
20
33
64
204
630
208
183
56
54
21
37
72
237
785
302
237
66
66
22
40
84
274
950
407
303
90
78
23
44
94
314
1155
562
381
105
93
24
48
108
358
1375
733
474
138
108
25
52
120
406
1646
974
582
159
126
26
56
136
458
1932
1238
708
201
144
27
61
150
514
2275
1597
852
231
166
28
65
169
575
2639
1987
1018
287
188
29
70
185
640
3073
2502
1206
325
214
30
75
206
710
3528
3058
1420
394
240
!
i
I
I
62
Table 1 - continued
I
I
1
1
2·
2
1
1
1
3
6
1
11
4
4
4
6
6
1
2
18
32
8
6
2
48
10
10
2
75
16
18
10
2
14
2
112
162
2
3
227
316
36
42
14
18
18
24
3
3
427
570
54
24
3
752
60
30
3
971
72
30
3
1246
84
4
1577
96
36
36
4
108
44
4
1977
2447.
128
44
4
3022
140
52
4
160
52
4
3676
4463
180
60
5
5357
I
2
2
i
24
30
I
,
1
63
Table 2 _ Number of (2n , 23) Designs
n
7
1
l'
8
1
1
1
1
2
2
1
1
4
2
2
2
1
7
4
6
2
1
3
1
11
19
3
4
2
29
2
44
2
67
2
97
9
10
11
1
13
1
2
3
6
14
1
5
10
15
3
9
16
8
10
16
6
18
22
14
17
11
28
32
18
5
6
18
18
45
44
22
7
2
138
19
20
32
66
59
28
8
2
195
49
99
77
34
2
270
21
76
137
100
40
9
10
3
366
22
115
191
126
48
11
3
494
23
168
256
158
56
13
3
654
24
238
344
194
66
14
3
859
25
334
445
238
76
16
1112
26
459
576
287
88
17
3
3
1430
27
619
729
345
100
19
3
1815
28
828
922
409
114
20
4
2297
29
1086
1141
484
128
4
2865
30
1414
1411
566
144
22
24
4
3563
12
i
Table 2 .. continued
n
II
3
1
1
4
3
4
5
6
10
6
7
22
43
8
12
21
34
9
55
132
10
11
83
122
215
337
12
13
176
249
513
762
14
15
342
467
16
17
624
826
1104
1571
2195
18
19
1075
1390
20
1771
7257
21
22
2249
9506
2813
12319
23
24
3507
4325
15826
20151
25
26
5313
6459
25464
31923
27
28
7837
9421
39760
49181
29
11296
60477
30
13439
73916
77
I
i
I
i
I
3021
4096
5486
\
CHAPTER III
CONNECTIONS BETWEF..N THE THEORY OF CONFOtJN1JED FACTORIAL DESIGNS
AND THE THEORY OF CODIm
1 - .Ill! theory of coding - Introductions .!:DS! definitions
The theory of coding is concerned with the development
of methods and techniques by which information can be transmitted
accurately and efficiently over a noisy channel.
We shall consider
a communication system, i.e•• one composed f1rst of a discrete
soyrce of information producing a continual sequence of discrete
symbols which are to be transmitted over a channel to a destination
point.
We assume the existence of noise in the channel which disturbs
the transmitted signal so that the signal received will in general
be different from the one stmt.
We say that an error of transmission
has occurred, in this case.
In order to minimize the effects of such noise; the
sequence of information symbols is divided into blocks of length k.
Each k-sequence, called a 'PUrC8 .8."ge, is then t!"ansformed in a
1:1 fashion by an encoder into a longer sequence of length n, called
a
code~.
The code word is transmitted over the channel by the
transmitter, to a receiver.
A d!cod,r determines from the received
word (if possible) the original source message.
In the above communication model we notice that an
66
n-sequence code word is used to carry a k-sequence source message.
The n-k
=
redundant symbols are used to correct some errors of
I'
transmission.
By properly choosing these
I'
redundant symbols for
each code word it is possible to correct or detect up to a certain
point the errors that occur during the transmission.
This is the
central problem of the theory or coding.
The set of code words is called a
code words of length n and redundancy
I'
.I!!k. and a C?ode having
= n-k is called a (n,k) .
code.
2 -
Add1~ye
errors
In the cOlllll1unication
mod~l
that we consider we shall
suppose that the noise causes additive errors only.
An additive
error is said to have occurred if the received word differs fran the
corresponding transmitted code word,
As an example, suppose that the 10-sequence
2£'
= (1
2 :3 1 2 0 0 1 2 3)
is sent and the 10-sequence
Zl
= (1
2 4 1 2 0 1 1 2 3)
is received.
The difference
between the two is the 10-sequence:
:l - 2£1 = (0 0 1 0 0 0 1 0 0 0)
where the operations
are mod. 5•. If we call the vector (Z - 2k) the error of transmission
vector or simply the error vector and denote it by
.§.
we have
We say that the received vector is the sum of the transmitted vector plus an error vector; hence the tem additive error.
The problem of correctiong additive errors consists
in dete.nnining the error vector 'Jl for a g1 ven received WQrd %'
the
67
transmitted word can t."len be reconstituted as
~
=:i. -.!!..
We now
introduce some definitions.
Let ~'= (xi' x2 ••••• x ) be a code word; we define the
n
weight of ~ denoted by w(~). as the number of non-null coordinates in
~.
If we define as before
a Z =(1 if a 1= 0
10 if a = 0
n
then w(~)
= r.
~
.
i=1
If ~,
= (y1'
.V2 , ••• , Yn)' we define the Hamming distance
between ~ and :i.' denoted by d(~, .l) as
i.e., d(~, .l) is the number of positions in which ~ and :I. differ.
The
Hamming distance satisfies the axioms of a matric.:
= 0 if A = :i.
= d(:i.. ~)
1)
d(~,:t.)
2)
d(~,:t.)
3)
d(M.l) ~ d(.A. li)
+ deli,
.l).
We now state a very important result, connecting the distance
property of' a code and its correcting ability:
if the Hamming
distance between any two code words is at least
d
= 2t
... v
+1
then it is possible to correct up to t additive errors and detect up
to t
+ v additive errors.
3 - Codes
~
A proof of this can be found in
1?
•
algebraic structure - Linear codes
Although this result
tells us what distance property
a code should have in order to correct and detect a certain number
68
of errors, it does not specify anything about the construction of a
code.
It is possible to study individually any kind of code, but to
unify the developnent of the theory we shall require that the code
possesses a certain mathematical structure.
A first requirement is that the code be a group under
t.~e
operation of addition, defined by
AI
+ ~1
= (xt
+ Yt'
~. + Y2' ••• , xn + Yn)
x2 ' ••• , x n ) and y' = (Yt' Y2' ••• , Yn) are code
words, with their coordinates as elements of a group. Such a code
where
AI
= (xi'
is called a group
~.
A more stringent requirement is that the code be a subspace of the vector space composed of all n-vectors with coordinates
in a Galois field GF( s).
A and
~
linear~.
If
are code words with coordinates in a Galois field we define
AI
and
Such a code is called a
CAl
+ ~I
= (xt
= (CX t '
+ Yt' x2 + Y2' ••• , xn + Yn )
cX2 ' ••• , cxn )
where c is an element of the field.
a group code.
We note that a linear code is
The converse is not true in general.
We shall now
confine our attention to linear codes.
Since a linear code is a vector space, all the code
words or code vectors can be generated fram a set of code vectors
forming a basis for the space.
If the code C is a subspace of
dimension k, a basis will consist of k independent code vectors
~t'
!2' ••• , !k'
We shall call such a code an (n, k) code over
GF(s), or a (n, k) code.
The k x n matrix G 1efined as
69
,-
G -
,
,
'
,]
[E.1 ' g2 ' ... ,
~
is called the generating matrix of the
(n,k) code C.
Each of the
sk code words .£ is of the fom
.£'=~'G
for some k-vector ~ with coordinates in GF(s).
An alternative way to specify an (n,k) linear code is by
an r x n matrix H of rank r = n - k such that
GH' = 0;
the code is then defined as the set of code vectors .£ such that
H.£
= O.
The matrix H is called the parity check matrix of the code C.
'j1e dual code CD of the code C is defined as the set of
n-vectors .£ such that
G.£ = Q.
or eq\l1va1ently as the set of vectors .£ of the fom
.£'
for some r-vector
= l.'B
1..'
with coefficients in GF( s).
We now relate the minimum distrance property of a code
that we introdu.ced in 2 with its structure in the case of linear
code.
First we have that
if the minimum distance of any two
words in a linear code is d then the weight of any non-null code
word is at least d.
This is true also for group code.
We can restate this in the case of linear code as follows:
a linear (n,k) code C with parity oheck matrix H as given in (J.2)
will correct up to t errors and detect up to t + v errors if and only
if H has property Pd-1 wher8 d = 2t + V + 1.
70
4 - Linear codes
~
factorial designs
Suppose given a (n,k) linear code C over the field
GF(s) with k x n generating matrix G that is C is the set of sk
code word s { £.
=~
I£.'
G,"!.'
= (}. 1'
). 2' .... ). k); ~ i C GF( s)
1•
~~en let us define the (sn. sk) confounded factorial design D
corresponding to C as the one with G as generatinp: matrix; that is the
set of confounded pencils of that design is
{
I
P<'~')
b'=~'G¥\'=(\
~
-J
A
~ l' ~2'
-
).. iE. GF( s)
•
In tems of algebraic structure we have on one hand a
linear space of dimension k generated by the k independent rows of
G; the s
k
elements of that space are the code words forming the code C;
on the other we have a (k-1)-flat of PG(n-1,s) generated by the k
independent rows of G; the (sk_1)/(s_1) elements of this flat determine
~~e
set of confounded pencils of the design D.
two code words
D two pencils
~
and £2 are identical if £1
P(~1)
and
P(~)
In the code C,
=£2;
in the designs
are identical when there exists an
element ~ i= 0 in GF( s) such that ~1
= J £2;
the null vector is not
allowed as coefficient of a pencil.
We could have also started from a (sn, sk) confounded
design D with a generating k x n matrix G and obtained the code C.
This shows that one can establish a one to one correspondence between
codes and factorial designs.
We shall now show that the important
problems in both theories correspond.
Let D be a (sn, sk) confounded design and C be a (n,k)
s-ary code both with the same generating k x n matrix G.
To every
71
confounded parallel pencil P(£') of D there correspond (s-1) code
worris of C given by j £' where j
in GF(s).
takes the (s-1) non-null values
Thus if the rlegrees of freedom confounded by
P(~')
belong
to a w-factors interaction this means that exactly w
coor~inates
2' are non null; thus the corresponding code words c
= ! ~'
of weight w.
of
are
This signifies that for a confounded design with no
(d-i) or less factors interaction confounded there corresponds a
code with the minimum weight at least equal to d.
statement is also true.
The converse
As it was shown in 3 we can translate
. the preceding results in terms of factorial designs:
for a (n,k)
s-ary code C which is able to correct up to t additive errors and
detect up to t + v additive errors there corresponds a (sn, sk)
confounded factorial design in which no (d-i) or less factors interaction is confounded; d
= 2t + v + 1.
Consider a factorial experiment in which each factor is
at s levels and where each block is of size sr.
An important problem
that was first studies by Fisher [9] and Bose [2] is to determine the
maximum number of factors which can be accomodated in such an
experiment so that no degree of freedom belonging to the main effect
or an interaction involving (d-1) or lesser number of factors is
confounded •. In view of previous results this problem is translated
in terms of coding theory as follows:
what is the maximum word length
n for which there exists a (n, n-r) code with minimum distance
d (r and d are fixer)?
Referring to section three,
be stated in algebraic language:
this problem can
given rand d one wants to know the
maximum n for which there exists an r x n matrix with property
pd-1.
vie often refer to the last formulation as the "packing problem."
72
Now that we have established the connection between the
theory of factorial designs and the theory of coding we can translate
easily the results of one into the language of the other and vice
versa.
In
[3]
Bose has stressed the connection between coding theory
and the theory of fractional factorial designs.
A fractional factorial experiment is one in which only
a fraction of the cOlllplete factorial experiment is used.
If the
cOlllplete experiment involves n factors each at s levels and we take
k
k
only a 1/s fraction of it we will say that we have a 1/s fractional
factorial experiment.
The way to choose this fraction is as follows:
we divide the total nUJllber of treatments in sk sets or blocks of
s n-k treatments each, the same way as we did for a(n
s , sk)'
experiment; i.e., the s
k
blocks are composed of the treatment
{X I G 1 = .1:; !' = (f1 ,
=(f1 , f 2 , ••• , f k ) takes
f 2 , ... , f k ), fiE GF(s)]
where,t'
a different value for each block.
We then
select one of the blocks as the fraction needed for the experiment;
i.e., we select the set of treatments
r
\1
,
GA
= ! 0 ; !0' = (f01 ,
(»)
0
f 20 , ••• , f k0) ' fiE
GF s
for a!0 vector.
It can be shown that the interactions confounded in the (sn, sk) de.1gns
remain confounded here.
Moreover, if the degrees of freedom carried
by P(~'), say, are unconfounded, these degrees of freedom will be
carried by E(&*')
also where
4.11,*' =
14' + L 1 G.
pel, I) and P(I£* I) are said to be alias pencils and the corresponding
interactions are called alias interactions.
The main problem here is the construction of fractional
factorial designs in which no t factors interaction is aliased with
73
another t-factors or lower interaction.
If
p(~')
carries degrees of
freedom belonging to a t factors interaction then we know that
wet)
=t
and in order that no t factors or lower interaction be aliased with it
we must have by (4.1) that
,
w(lt·)
or equivalently
4.2
= welt'
+ l.'G)
,
d(1' •.!.'G) ~ t
= d(lt' ,-~'G) ).
t
¥,,!. I:
0
¥2.1: o.
A.s we have the inequality d(li', A.'G) ~ d(Q., ~'G) - d(Q., 1')
.for all ~ we can rewr1te (4.2) as
d(Q, l.'G) ~ 2t.
Hence this means that in the (sn, sk) confounded factorial design
k
used to construct the 1/s fraction we must have that the lowest
interaction confounded has at least 2t + 1 factors.
We summarize
'--
this as follows:
If a (sn, sk) confounded factorial designs so that
all 2t or lower interactions remain unconfounded we can construct from
it a (n,k) s-ary code capable of correcting t additive errors and
we can deduce a fractional 1/sk factorial experiment in which no
t factors interaction 1.s aliased with another t-factors or lower
interaction.
The eJdstence of any of the above codes or designs
implies the existence of the other two.
5 - Perini tions .2f Qptimalitv
We have seen above how the theory of factorial designs,
the theory of codes and the theory of fractional factorial designs
are closely related and how the problems of one theory can be translated into the language of the others.
These connections between the
74
three theories result from the fact that in each case we study the
set of e18lllents
C ={~
5.1
I ~t =!..'G.
>.
= (~1' ~
2' ....
X k)'}
~ i4; GF(s)
or equivalently
5.2
C
={~
IQ=
H
~.
G H'
= oJ
where G is called the generating matrix.
In the context of the theory
of designs we consider C with a structure of Euclidian geClfletry
defined on it; in the context of the theory of codes we consider C
. as a linear space.
In order to avoid the repetition of the same thing in
three different languages it is more simple fl'(lfl now on to talk only
of the set C with generating matrix G. as defined in (5.1).
'!he
algebraic structure underlying the set C will be specified as we go
on.
The important parameters in the study of Care s the
number of elements of the basic field GF( s). n and k which are the
dimensions of G. and d the minimum weight of the elements of C.
Given s we can fix two parameters and optimize the third one to
obtain the three definitions of optimality:
fixed parautters
Qptimization
types of set C
n.k
maximize d
maxi-min distance
n.d
maximize k
maximum
k.d
min:1.m1 ze n
minimum redundancy
size
For the case k = 2 we shall study now the set C am find
the one with maxi-min distance and the one with minimum redundancy
as defined above.
In order to study the case when k = 2 we will use
75
the results derived in the section 4 of the last chapter.
The generating matrix G is a 2 x n matrix.
roe 1, s)
column of G represents a point of
Each non-null
and we obtain using that
representation. the generating graph corresponding to G.
An example
in the context of the confounded factorial design is givan in section
4 of Chapter II. Another way to look at this question is to study
the modular representation of CI let us denote the (s+1) points of
PG(1,s) as P1' P2 , •••• Ps +1 and let us define G as the number of
i
columns of G representing Pi for 1 = 1, 2, ... , (s+1) and GO as the
....1' .
-number of null columns of G. We ::rne ~ I : (G1 , G2 ••• , Gs +1 ) as
s:
the modular representation of C;
G
.
1=1 i
=n
- GO.
If we say that two sets C1 and C are similar whenever
2
the corresponding confounded designs are similar, we can SU1lIlll8.rize
the results of the section 4 of the last chapter in the following:
l'heorem
.2a.!
If C is a set as defined in 5.1 with generating matrix
i
G and with the ·following modular representation
1
,
s+1
Pi
(Gi ' Gi ' ••• , G1 s +1), ~
Gi
n - GO' i
1,2
1
2
J=1
j
then C is similar to C if and onl.v if there is a (s+1) x (s+1)
1
2
=
=
=
pe.nl1utation matrix Es +1 for which we have
Es +1 G1 = G2
1. e. G1 and G represent the saine partition of n - GO.
2
Let· us now recall the example of section 4.
matrix is
11 0 x x x 0 x)
G 1.0 1 x 0 0 x x
the generating graph in Rl(1,s) is
•
The generating
76
,
(1,4,5)
,
,
(2,6)
(3)
'.
where the numbers stand for the colUJllns of G.
,
One modular representation of the set C associated with
G is ~ = C3,2,1,1,O,O, ••• ,0). (The number of seros in ~ . depends on
the value
I.)
If in C we do not distinguish between
~ = ~:I.
'I:
(~
~
and .I wheneftr
0 an element of G,( s» and if we do not include .Q.
in C then C contains s+1 elements.
In the present example the
,weights of the elements. are as follows:
weight
nUlbtr of elsm,
4
1
5
1
6
2
7
8-3
In general for a set C with modultr representation
.
s+
~I
II:
~
(G1' G2 , G , •• Of Gs+1 ).
3
Gi = n - GO
the 8+1 e181llent8 have their weights equal to
8+1
5.:3
L
Gi - Gj = n - GO - Gj
j
II:
1, 2, ... , s+1.
i=1
We then
have the
For fixed n and k = 2 the problem to tind a .et C with
maxi-mini distance is equivalent to choosing GO and
~.uch
that
s+1
(1)
L
Gi
=n
- GO' GO ~ 0 and Gi ~ 0) i
i=1
(2)
j=1~~s+1 (n - GO - Gj ) is maximised.
II:
1, 2, ... ,-.+l
77
A first step in solving this problem is to put GO = 0
for
~(n-Go-Gj) ~ ~(n-Gj) =n-maxG j ;
me
then we want to find
~
-
(n - max Gj ) = n - II1n max Gj ;
~
j
j
or equivalently we want to tind
s+1
ml.n
~
L
tor ~ satisfying
Gj
JI1IlX
Gj = n.
i~
j
The solution of this problem is the following,
it n
=k(s+1)
let Gi
it n = k( 3+1) - r
G1 = G2
j
for 1
r
=... = Gr = (k-t)
in that case min max G. =
t:
=k, i = 1,2,
s
••• , n;
let
and Gr+l
[n/(s+l»)
+ 1
= ... = Gs +1 = k.
(where
(s]
is the
J
largest integer smaller than s) and the maxi-min distance is
,.4
d
max
=n
( n/( s+1 )J
-
- 1.
In order to solve the minimum redundancy problem we state
the obvious.
For fixed d and k = 2 the problem to find a set C with
~
minimum redundancy is equivalent to choose GO and.!i
(1)
min (n - GO - Gj ) = d
j
such that
= 1, 2, ••• , s+1
s+1
(2)
GO +
1.
Gi = n
is minimized.
i=1
In a first step to solve this problem we can set GO = 0,
s+1
then we need to minimize n = ~ Gi when
d
= min
j
i=1
(n - Gj)
=n
.. m~ Gj
J
78
or equivalently we want to find the
,...
min n = d + min max G for a proper choice of 2,.
N
I'W
~
Q.
j
j
A solution to that is to take the set C with maxi-min distance d
constructed above.
Notice:
In the construction of the maxi-min distance
set C, we have supposed that GO = O.
This is not necessary.
For
example, in the binary case the following sets C with
and
modular vector
GO value
(4,4,4)
1
(5,4.4)
0
(5,5,3)
0
where n = 14, k = 2, s = 2
are all of maxi-mini distance 8.
The first of these sets is not of
minimum redundancy because GO = 1.
what we have said.
This is not in contradiction with
We have proved only that there exists a set with
maxi-mini distance which is also with minimum redundancy and this
set has GO = O.
We now translate these results into coding and design terminology as follows:
di'stance d = n-1that distance.
no {n-2-
there exists a (n,2) 5-ary code with maxi-min
[n!(s+O]
which is of minimum redundancy for
There exists a(sn, s2) confounded design in which
(n/( s+1») }
-factors interaction is confounded.
There
exists a fractional 1!sk factorial experiment in which no
t
= ~ (n-2- l
n/(s+1)]
)!2} factors interaction is aliased with
another t-factors or lower interaction.
Let us now work out an example which will illustrate the
79
connection between the three theories.
Let us consider a set C with generating matrix
G = (1 a 1 1 0 2 1 2 10)
\0 1 2 1 1 1 0 2
here n = 9, k = 2, s = 3
and the generating graph may be represented as
P1=(1,O)
P =(0,1)
,
,
2
(1,7,9)
(2,5)
P =(1,1)
3
P4=(1,2)
•
•
( 4,a)
(3,6)
"YNe notice that this set C is of maxi-min distance 9-1- [ 9/ 4 ]
=6
and for that distance C is also of minimal redundancy.
The (39 , )2) confounded factorial design obtained from C is composed
of the following 9 blocks of )7 treatments each
{9!1 \
2G9~1
= .,y; :I.' = (yl' Y2)
Yi = 0, 1, 2;
i = 1,2} •
The confounding structure of this design is the following:
F F F F F Fa F
1 3 4 6 7
9
F2 F F4 F F6 Fa
3
5
F1 F F4 F F Fa F
2
9
s 7
F1 F2 F) F F6 F F
7 9
s
As we know the smallest order of the confounded interaction is 6 and
this is the best result we can obtain.
The (9,2) 3-anv code corresponding to C is composed of the
words
80
101 102 121
o1
2 1 1 1 020
1 1 021 0 1 1 1
1 2 2 021 101
2 0 2 2 0 1 212
o2
1 222 0 1 0
220 1 2 0 2 2 2
2 1 1 0 1 2 2 0 2
o0
0 0 0 0 000 ,
. the minimum distance is 6 and for that distance the code is of
minimal redundancy. One is able to correct with this code up to
two errors and detect up to three errors.
The 1/9 fraction
2!: !h!
)9 factorial design associated with C is
composed of the set of treatments
{A \ G X = Yo
Yo
= (y10' Y20)} •
We know that no two factors interactions will be aliased
with another two or lower order interactions and in such case this is
the best that can be done.
6 - Optimality in the case s
=2, k =)
In the last chapter we studied the problem. of enumeration
of (2 n , 2)
confounded factorial designs.
We may use here these
results to study the optimality of the sets C defined as
\1 \ l'
=
1.'
G,
!,
1:
(~1' A 2'
} );
~
ita GF(2) for}
i = 1, 2, )
. where G is ) x n matrix of rank ) with entries in Gr(2).
We may
consider the non-null columns of G as representing points in PG(2,2).
81
The geometry PG(2,2) consists of 7 points and 7 lines.
representation of this geometry is given in Figure 6.1.
as before
t~e
A graphical
\V'e denote
modular vector
(1,0,0)
corresponding to the generating
= P2
matrix G where Gi is the
number of columns of G which
represent the point Pi
i = 1, 2, ••• , 7.
We define
Go as the number of null
( 0, 0 , 1 ) =P1 . (0, 1 , 1 ) =P'3
( 0, 1 , 0) =p4
columns of G.
Figure 6.1
We have GO + ~ Gi = n. C considered as a vector space or a projeci=1
ti ve geometry has seven non-null elements whose weights are given
by (See (5.1) in Chapter 2)
G2+i + G +i + G6+i + G7+i
i = 1, 2. 3•••• , 7
5
where the index 8 = 1, 9 = 2, etc. •••• (6.1) can be written also
6.1
as
n - [G 1+i + G +i + G4+1] - GO i = 1, 2, 3, ••• ,7
3
and the minimum weight for C is
6.2
6.3
t LG1+i
n - m:uc
,...
+ G'3+i + G4+i1
1·
+ Go
In order to obtain the maxi-min distance set C we must find
~
-
Q which maximize (6.3) which is equivalent to finding
6.4
where
min max {(G1+i + G +i + G4+i) + Go}
3
~ i
7
GO +
L
i
=1
Gi = n.
82
TIle first step in solving (6.4) is to set GO
= O.
the choice of ~ which will solve the prabl. is ~'
and we prove that as follows:
~'
for any
If n
= 7k
= (k,k,k,k,k,k,k)
vector the sum of the
weights in C equals 2&; the maxi-min distance (or weight) 4k
is obtained when all the weights equal 4 k and this occurs when
G
i
=k
for all k.
It n = 7k +1 an optimum -choice tor ~ is to take Gi ~ It
?
f'or all i and ~ G
i
= 7k + 1.
In this case the optimum
~
~
is not
The proof' is iJimilar to the one given in the preVious caee.
unique.
If' n
= 7k
,...
+ 2 an OptimlDll choice tor
?
Gi ~ k for all i and ~ Gi
= ?k + 2.
~
is to take
The minimum distanoe is 4k in
1
this case.
If n
~'
the
IIlin:1DlUlll
optimum
~
= ?k
=(k+1,
+ :3. an optimum choice is to take
k+1, k+l, k,k,k,k);
In general we will have an
distance equals 4k+l here.
if the three Gi with value k+l correspond to three non-
colinear points.
If n
~'
= ?k. + 4
= (k+l,
an optimum choice is to take
k+l, k+l, k, 1&:+1, k, k);
In general we will have an
the minimum weight equals 4k+2 here.
optimum
~
if' the f'our G with k+l value correspam to four points,
i
no three of them colinear.
If n = 7k + 5 the optimum roice tor
k+l ~ G ~ k
i
i = 1, 2, ... , ? ard
IIlin\Jaum weight equals 4k + 2.
If n
= ?k + 6 the
I
G
i
= n.
~
is to take
In this case the
1
optimum ohoice f'or ~ is tD· take
83
7
"---
k+l ~ Gi ~ k i
= 1,
2, ••• , 7 and
I:
Gi
= n.
In this case the
1
minimum weight equals 4k + 3.
We summarize the results in the following table:
n
maxi-min di stance
7k
4k
k ~ 1
7k+l
4k
k ~ 1
7k+2
4k
k ~ 1
7k+3
4k+l
k ~ 0
7k+4
4k+2
k
7k+5
4k+2
k ~ 0
7k+6
4k+3
k ~ 0
~
0
All the results above can be translated easily in terms of designs or
in terms of coding.
7 - The dimensions of the sets C
In the preceding sections we have translated the design
problem into coding terminology and vice versa.
However, when it
comes time to consider the realization of these theories we must
realize that a useful confounded factorial design does not usually
correspond to a
use~ll
linear code and vice
versa~
This does not depreciate at all the theoretical connection between
these fields but we have to realize that the examples given above
serve only an illustrative purpose.
CHAPTER IV
HEIG!:tT DISTRIBUTION OF LINEAR CODES
1 - Introduction
The performanco of a linear code depends on many parameters such as the lenr,th of the code words, the number of information
places, the minimum weight of the code words and the weight nistribution of the code.
We would like here to study more particularly
the weight distribution of the code.
Some investigations have already
been done in [18J for example where with the aid of a computer one
obtains by enumeration the weight distribution.
cases, formulae were obtained as in [4],. tF],
For a few particular
t8l,
[11], r13], [16] •
In this chapter we will study the problem of weight
distribution by studying the algebraic structure of the parity check
matrix of a linear code.
In particular we will show that the weight
distribution is a function only of the quantities NU , the number of
v
v columns of the parity check matrix with rank u. This is applied
to obtain a new formula for the weight distribution of
t..~e
distance
three s-ary Hamming code and a formula for the weight distribution of
2
2
the distance four (s +1, s -J) s-ary aonic
codes.
For s
= 3,
4, 5,
7, 8 we have calculated the weight distribution of the distance four
conic
code and we give the results at the end of this chapter.
Finally we have derived an upper and lower bound for the weight
85
distribution of a linear (n,k) code.
2 - Notations
.!nsi definitions
We consider a (n,k) code C over a s-ary channel.
Such
code can be nescribed as the set of n-vectors (code words)
AI = (~, ~, ••• , x n ) such that F-l
matrix of rank k, xi&' GF(s).
of the code.
= 0, where
H is a n x (n-k)
We call H the parity check matrix
TI1e weight of a n-vector is defined as the number of
non-null coefficients of the vector.
The minimum weight of a code
.C is the smallest integer for which non-null code word with that
weight exists.
We denote the minimum weight by d.
the number of code words with weight i.
The set
A(i) will denote
A(i); i
= O,1.2•••• n
is the spectrum or the weight distribution of the code.
For a (n.k) linear code C with minimum weight d we
want to determine the spectrum of C.
k
exists (s -1) non-null code words ~'
We already know that there
= (xi'
x , ••• , x n ) , ~ C GF( s)
2
each of weight d or more.
First formula for the spectrum
Denote by E the event that the i th coefficient of a
i
code word is null.
There are n such events E , i = 1, 2, ••• , n.
i
m
n ... n
Denote by N( E
E. ) or N( .~1 E ) the number of times
i1
ij
~m
Jthe event ~1 E j occurs or in other words, the number of code words
i
with the iih , •••• i~h coordinates null.
~enote by Nt the number of
code words with exactly t of their coefficients null.
(1 )
Obviously
Nt = A(n - t).
But Nt is the number of times exactly t among the events
together.
E
i
occur
The M8bius inversion fonn.ula relates the quantities Nt
86
m
and N(jQ1 i 1 ) as tollows:
j
n
(2)
Nt =
r
(_1)m-t (~)SJIl tor t ,
m=t
n-d
L. ... i.
where S =
w( m E )
1
m l ' 1 1t 1 2(. ...~ iJll."
j=l j
tor the case 0 L (n-t) '- d we have Nt
=0
and No = 1 because the
code C has min1m\Dll distance d.
TheJllain work is now to evaluate the quantities S. Le't
m.
a
us notice tirst that N(
1 ) 1s the nlBber of solut10ns 1 of ~ • 0
.
j-1 1 j
th
th th
.
tor which the 1 1 ' 1 2 ' ••• , i m coett1aient ot 1 = (lit' ~ t • • • • ~)
n
m
n
E ) 1s the nUlllber ot solutions of a syst. of
j=1 1 j
(n-k) -= r equations in n-m unknowns and depends on the !'ank of the
are null.
N(
system ot equations.
In a code C with min1mUlll Weight d, the parity check
matrix H ot the code has prope:rty Pd-l; that is, no (d-t) colUIIIUJ of
H ue linea!'ly dependent.
system
HI
=0
Thus 1t n-m £ d the only solution to the
th
th
til
tor which the 1 1 ' i 2 t m••• t i ll1 coordiDates &Jte
is a null vector (0, 0, ... , 0):
N(
n Ii ) = 1 to!'
j=1
S = (n)
m
m
we rewrite
~
n-d and
j
to!' m .) n-d.
equation (2) as
n
n-d
(4)
III
Hl'O
Nt
=1
(_1)m-t (:) Sa +
m=t
1:
m=n-d+1
(.1 )m-t
(:> c:>,
tor t .. D-d
The tol'lllula (4) is valid tor any (n,k) s-a!'y linear code
with JIlinimUlll weight d.
87
Optimal
~
We can apply the formula (4) to the case of "optimal" code.
We say that a code is optimal when d-l = r.
In such a case
m
N(
.n
J=1
Ei .) is the set of solutions of a system of r equations in
J
n-m unknowns and the rank of the system is r when n-m
~
r (because
H has property Pd- 1 ) • Hence there are s k-m vectors solutions of
such a system
(5)
s
k-m
for n-m
~
r = n-k.
The equation (4) becomes in this case of optimal codes
k-l
(6)
Nt =
L
n
(_1)JI1-t (~) (~) sk-m
m=t
+!-
(_1)m-t (~) (~)
m=k
for t !: k-l.
Using the identity
(7)
(m) (n)
t
m
= (n) (n-t)
t
m-t
we can rewrite (6) as
(8)
Nt =
(~)t
k-l
L
(_1)m-t
(~:~)
n
sk-m +
~t
or as
Nt =
(~){
L
m~
n-k
k-t-1
1.
(_1)m (n;t) sk-m-t +
m=O
2: (-1)m-t-tk(m~t~)J.
m=O
This result on optimal code was found by three groups:
E. F. Assmus, A. M. Glesson, H. F. Mattson in [1] , T. Kasami, S. Lin,
W. W. Peterson in [14] and
[15] , G. D. Forney, Jr. in (101
From this formula we will make two observations:
Lemma 1
Ad ~ 0 for all optimal codes.
88
Proof:
~-1
Ad = Nn- d = Nn-r- 1 = Nk - 1
n-k
(k~1) [s
=
+
L
(_1)m+1
(n;~r1)]
m=O
n-k
m+1 (n-k+1)
m+1
= 0 we have that
as ~ ( -1 )
m=-1
~-1 = (k~1) (~-1) ~ 0 for s ~ 1.
'Ibis means that in an optimal oode the minilftuiD weight 18 attained.
~
For r
Proof:
A .I: O.
s and n = r + 2
n
A = N
= N
n
n-n
0
k-i
NO = ~
n-k
(_1)m (:) s
k-m
~
+~
m=O
m=O
n-k
Using the identity ~
(_1)m ( n ) = ( ... 1)n-k en-i) and the raot that
m...k
~
n...k
m=O
k= n-4 = 2 we get that
NO
= 82
and as funotion of
- (r+2) 8 + r + 1
St
= (8-1)
(s-r-1)
NO looks like Figure 1 henoe for
r
~ S ~
1 we have NO L O.
The oodes with these parameters·
do not exist•
•
Figure 1
Corollary I
n ~ r
+ 2.
'!here are no optimal oodes with r
~'
sand
i~
</'
89
=r
The case n
+ 2 is obvious from Lemma 2.
~
a code with n ). r + 2 .> r
If there is
s then we take r + 2 columns of its
parity check matrix as the parity check matrix of a new (r + 2, 2)
optimal code with r
~
s
whiCh
is impossible.
The last observations were made by R. C. Singleton in [191
using a different argument.
3 - Second formula for the spectrum
As we noticed earlier the main problem in the evaluation
~f
Sm
the spectrum of a code is to know the values S •
m
=
m
L·.. L
N(
1 L- i 1L. ••• J. i m, n
of solutions
of
A of HI =
.Pi
We recall that
m
Ei
)
where N(
j
0 for which the i
A are null. If we form a new matrix
n
E .)
i
j=1
J
is the number
th
th
th
' i 2 ' ••• , i m coordinates
1
Hi by deleting the
th
th
th
m
i 1 ' i 2 ' ••• , i
columns of H then N( 1\ E ) is the number of
m
j=1 ij
(n-m)-vectors X. solutions of the equations Hi!.
0 and depends on
=
the rank of Hi.
m
N( (\
j=1
)
E
i
j
If the rank of Hi is u (r ~ u ~ d)
= s n-m-u•
Let us denote by N~ the number of r x v su1:::matrices Hi
formed with v columsn of H such that the rank of Hi equalsu.
Obviously min (r,v) ~ u ~ d-1
for v ~ d-1, and we can write
min(n-m,r)
(10)
Sm
u
n-m-u
s
for n-m
n-m
= L
N
~
d
u=d-1
and from (4) we have S
m
= (n) for n-m
m
L
d.
We rewrite (4) as
90
min(n-m,r)
n-d
(11 )
(_1)m-t (~)
L
Nt =
m=t
n
u=d-1
(-1 )m-t (~) (~)
L
+
NUn-m s n-m-u
L
for t "
n-d.
m=n-d 1
u
The last formula depends only on N which are easily
v
known in certain cases.
On error detecting Hamming codes
As a first illustration of formula (11) jwe look at the
,well known family of one error correcting Hamming codes.
for computing their spectrum is given in
[17].
A method
It is shown that the
number of code words of weight j in a s-ary Hamming code is found as
the coefficient of x
j
in the following polynomial
n:1
( 12)
f(x) =
n( s-1 )+1
1
[ (1+(s-1)x)n + n(s-1)(1+(s-1)x) s (l-x)
n( s-1 )+1
with n
s
J
= (sm_ 1 )/(s_1)
which becomes for the binary case
...L
n:1
f(x) = n+1 [(1+x)n + n(1+x) 2
(13)
n+1
(1-x) 2
J with n = ZU-1.
Let us look at the s-ary one error correcting Hamming
code as a cyclic code.
H
an
ai
=1
[
..I
~ ~
I
Its panty check matrix is
;t
2 :'
~
..I
t
nJ
... ;ci
m/
with n = (s -1) (s-1)
is an m x 1 column vector which corresponds to a point of
PG(m-1, s).
In projective geometry language N~ is interpreted as the
number of sets of v points of PG(m-1, s) that span a (u-1)-fiat.
u
We now proceed to calculate the quantity N that we shall denote
v
N~(m-l, s) because we are working in PG(m-1, s).
91
Result 1N11(m-1,s) = n and Nt1 (m-1,s) = 0 for t ~ 1,
Proof: N1(m-1,s) is thenumber of sets of t points
t
which generate a O-flat and the result is obvious.
Result 2.2(
1 ) _(f(m-1,o,s»
N m- ,5 2
2
~(m-1,s) = (S~1) f(m-1,1,s).
Proof:
~(m-1,s) is interpreted as the number of sets of
two points which generate a 1 flat.
As any pair
of points generates a 1 flat (a line) we obtain
To obtain Nt2 (m-1,s) we count the number
of lines of PG(m-1,s) for each of which there are
the result.
( s+1)
t
sets of t points.
Note:
)(N)
(N-m+1 )
s -1 ... s
-1 1. (N,m, s) is the
'1'(N,m,s) =
)"
( sm -1 )( sm-1 )... ( s-1 )
f.
\s
..i\
N+1
-1
+1
number of m-flat in PG(N,s).
Result 3Ntt (m-1,s)
=
~ (m-1, 0, s)
tI
t-2
,1
11 {i(m-1,o,s) - !(j,O,s)J
j=O
t ~1.
Proof.;
Nt (m-1,s) is the number of ways we can choose
t
t independent points in PG(m-1,s) and the result
is well known.
Results 1 and 2 are particular
cases of this one.
Result 4-
N~(m-1,s) = ~ (m-1,2,s)
Proof:
{(
t (2,~,1~ - ~(2,S)}
N,(m-1,s) is the number of sets of 4 points which
92
generate a 2-flat and we count these as follows:
F~r a given 2-flat there are
(f
(2,0,1)\ sets
1
4 J
of 4 points out of which N (2,s) generate a 1
t
flat the difference being the number of sets of
4 points which span the 2 flat under consideration.
We multiply by f>(m-1,2,s) (the number of 2-flat
in PG(m-1,s»
Result
to obtain the result.
5Nf(m-1,s) = f(m-1,2,s)
{(4)(2,~,1))
-
~(2,S)}
•
This is the generalization of Result 4.
Result 6-
N~ (m-l.S~ ~
Proof:
!(m-l.J,s)
{(+(J.~.1l)
- N;(J.S) - ~(),s)}
N (m-1,s) is the number of sets of 5 points which
5
span a 3-flat (i. e. 4 of which are independent).
In each 3-flat of PG(m-1,s) there are
sets of 5 points out of which
a 2-flats and
(fC3,~,S»)
N~(3,s) generate
~(3,S) generate a 1-fiat. The
difference is the number of sets of
5 points with
four of them independent. Multiplying this by the
number of 3-fiat in PG(m-1, s) gives us the result.
Result
7-
N~(m-1,s) = ~ (m-1,u-1,s) {(4)
1
1L
uLv~n
N~(m-1,s) = ~(m-1,O,s)
Proof:
u-2
(U-1;O,S»)
-
or N~-f(U-1,sry
..f =1
t-2
t~(m-1,O,s) - ~(j,O,s~)t ~1.
tI
jaO
The first part is a generalization of the preceeding
results.
11'
The second part is a repetition of
r~t
1
93
Having a knowledge of NU we are now in position to compute
v
(11) and to calculate the spectrum of the code.
In the binary case the method using (13) is much simpler
than the method suggested here:
we can easily derive fram (13) that
A(j)
=nh{(~)+.n(-1)j/2«n-~J~2»)
A(j)
=-1[en)
n+l
j
for j = 2k
+ n(_1)(j+1)/2«n-l)//2)] for j = 2k + 1
(j-l) 2
and the properties that
An = 1
A _1 = A _2 = 0 Aj = A _ j for j = 3.4, ... , (n-7)/2.
n
n
n
In the non-binary case however the method suggested here
offers an alternative for the method using (11).
The derivation of the quanity NU has also a combinatorial
v
interest by itself.
~
one error correcting -
~
error detecting
~
conic codes - r = 4
The parity check matrix H of an s-ary one error correcting two error detecting conic code has property P3; that is no three
colUlllns of H are dependent.
For the case r = 4 we may identify a
column of H with a point of PGO,s) and in that projective geometry
language H has property P if no three points represented by the
3
columns of Hare colinear. It has been shown that the maximUlll number
of points we can choose in FG(3,s) with no three of them collinear is
s2 + 1 J (s) 2).
There are the s2 + 1 points on the non-degenerate
qUadratic surface S defined by
(15)
a
~ + 2h Xo Xl + b
xi = ~ ~
a,b,h GF(s).
Consider P1 and P2' two points on S and P1P2 the line joining P1
and P2.
By P P there pass (5+1) planes.
1 2
It can be shown that each
of these planes contains exactly (s-l) points of S excluding Pl
and P2.
NU can now be interpreted as the number of sets of v
v
points of S lying in a space of dimension u.
N5
2
= (s ;1) is the number of sets of three points on a
plane.
2
~1) (Si 1 ) (s+1)/(~)
N' = (s
is the number of set of four
2+1
(S 2 ) is the number of Pairs of points in S•
points on a plane.
.( s+l) is the number of planes passing through each pair of points.
(Si 1 ) is the number of 4 coplanar points we can form with the original
4
(2) is a multiplicity factor.
2
4
.. N3
N4 -_ (s +1)
4
- 4
pair.
and in general
2
(16)
Nf = (s
(17)
4
Nt
~1) (t~) (s+l)/(~)
=(s2+1
t)
3
- Nt·
Formulae (16) and (17) are the ones we need for computing the spectrum
frCll1 (11) which becomes
2
N = s~3 (_1)m-t(m){N3
(s-1) + (i+1)l
t
L
t
2
+1
m J
s -m
.
m-=t
2
s s -m-3
(18)
Formula (18) may be calculated with the aid of a cCll1puter when
calculation in integer of large value is available.
Indeed one
obstacle to the computation is the presence in the formulae of
quantities as ( tn) or s n-m-t which may become very I arge for medium
95
values of n (25 to 60).
4 s-ary
~amming
The weight distribution of the distance
codes has been calculated for s =
3. 4. 5. 7, 8
I.
and are produced in Appendix
4 - Bounds 5m A(V)
We now consider a s-ary (n,k) linear code with minimum
word weight d (d
(4)
4:
n-k
= r).
n-d
Nt
t
=L
(_1)m-
We recall fomula (4)
n
L
(~) Sm +
m=t
m:n-d+1
for t
m
where Sm
" n- d
I ,,·L
=
Nt is the number of words of weight equal to n-t 1. e ••
Nt = A(n-t).
\Vhat we intend to do here is to derive two quantities
S- and S+ such that S- '- S ~
m
m
m-m
S+(m L. n-d) and use it in (4) to
m-
obtain bound on Nt.
Perivation
.2!
S +
m
m
N(" E ) is the number of solutions of HI
0 where
i
j=1
j
rHm is the parity check matrix of the code and the coefficients
=
~ ,Xi ••••
,xi of 2S are zero.
12m
th
th
If we fom the r x (n-m) matrix
th
' ••• , i
Hi by deleting the i 1 ' i
m columns of H. it is easily
2
m
seen that N(.l\
J;i
E
i
,>
is the number of solutions of H1l1 = 0
J
(At a (n-m)-vector) which depends only on the rank of H1 • Sm is the
summation over all possible r x (n-m) such matrix Hi fomed by deleting
m columns of H.
In order to get an upper bound on Sm .we will suppose that
H is a parity check matrix of the fom [I r
ll\1
such that the rank
96
-
of any submatrix of m columns or more is at least d (m ~ d - 1).
However from the structure of
:r
it is obvious that not all subnatrices
fonned of m columsn are of rank d.
As an example ,if m1 of the first
r columns of H are included in H1 , its rank is at least mi.
~
r
m1
~
If
d then H1 has rank mi and the system HiAi = 0 has at most
n-m-rn i solutions.
S
By enumerating all posible subnatrices Hi formed with
n-m columns of H we find that
S+ = ( r ) + ( r 1)(n- r) s + ••• + (dr ) ( n-r ) sn-m-d
1
m
n-m
n-mn-m- d
(19)
+
rL (mn )
,
(
_ n-m-d
L
r )(n-r) ]
n-m-i
i
sn-m-d+1
if r~ n-m ~ d
i=O
where (n~m) is the number of r x (n-m) matrices Hi formed with (n-m)
of the first r colUll1n8 of H =
lI
:
M1 ; (n-~-1)
r
(ni )
is the number
of matrices Hi formed with (n-m-1) of the first r columns of H and
a of the last n-r columns of H.
Such matrix Hi has minimum rank equal
to n-m-l and H1Al = 0 has at most s solutions.
(19) are similarly calculated.
The other terms of
We rewrite (19) in a concise form
(20)
r ~
n-m ~ d.
For other values of m we obtain
r-d
(21) S+ = (n) sn-m-d+1 _ ~ (r)(n-r) s n-m (-d+i
s
-s-i-4)
m
m
~
i ~i
i=O
if n-r+d ~ n-m ~ r.
r
(22)
S: =
I.
(~) (~:~) sn-r-m-i if n-m
i=O
Derivation of Sm
> n-r+d.
97
In order to get a lower bound we will suppose that
! 1
~'\: and all subrr:atrices Hi fonned from l: in the usual
r
way atta;.l their maximum passible rank. By a similar enumeration of
H = [I
the matrices Hi as we did for deriving fonnula (19) we get now
(23)
s-m = (n)
m
(24)
s-m
for r
= (n) s n-m-r
~
m
n-m
n-m~
~ d
r.
These bounds on Sm are quite crude because we only used the minimum
infonnation possible. When more infonnations are available about
H ,these
bounds can be improved.
vie can utilize fonnulae (20) to (24) in order to find
bounds on Nt as follows:
In equation (4) replace S by
m
replace Sm by
S- for m-t odd
m
S+
m for m-t even.
The resulting expression noted N~ is an upper bound for Nt.
On the other hand replace Sm by
replace S by
m
S+
for m-t odd
m
S· for m-t even
m
and the resulting expression noted N~ is a lower bound for Nt.
1-le
have then Nt ~ Nt ~ N~.
Another upper bound for Nt
G. D. Forney in [10] derives a very simple upper bound for
+
Nt which we denote by Nt
+
to'
(25)
Nt (n.t). (t+t )I
o
where to is the largest integer such that 2tO '- d.
n,
derive (25) is very simple and goes as follows:
The argument to
the total number of
words of weight n-t-t o distance to from a code word of weight n-t
is (ntt)
a
since to eet such a word we may c!1anfs any to of the n-t non-
null symbols in the word to zeroes.
Hence the total number of words
98
of weight n-t-t
o distance to from some code word of weight n-t is
(~-t)An_t and all are distinct, but this number cannot exceed the total
o of vectors of weight n-t-t equal to
number
o
+
It is difficult to compare Ntand
o
~
Nt
primarily because the
Let us however study the behavior of these two
limits at t = (n-d).
(26)
(27)
)( s-1 )n-t-t0
(25).
which gives formula
complexity of Nt.
(n-t-t
n
We have
N~_d
= [(~) -
~+
= n'
(~)J (s-1)
to'
(s_1)d-to
n-d
dl (d+t )'
•
O
+ ~
In case of a binary code we have N d > it-d iff
n-
n'
....nl.- -
rl
<.
to'
(n-d}l
(r-dH > (d+tO" •
The last expression being difficult to visualize let us replace the
(28)
factorial numbers by their sterling approximation:
(29)
t d [<n_d)d
_
(28) becomes now
(r-d)dJ ~ n'
+
N:+
nn:+
n.I:+
nl and Nn- d~ Nn- d'
which is an approximate criterion to compare N d and N d'
that as n increases we will have n
Also from
d
<
We see
(26) and (27) we have that N:_d"'~:_d when s gets
large and other parameters stay unchanged.
This comparison of the behavior of the two upper bounds is
very restricted.
It shows that it may be advantageous to compute both
n.I+
upper bounds and take the smallest. The Nt bound has the great advantage
+ bounds may often give a lower value.
of simplicity; however, the Nt
I want to mention the recent paper of J. E. Levy:
weight distribution bound for linear codes"
"A
IEEE, IT-14, No.3, p. Qat
99
5 - Future research
This work is made around the general problem of enumerating
the different types of (sn, sk) factorial designs.
If we have
solved some particular cases, the general problem remains unsolved.
One could get some more insight into the problem by studying the
case of the enumeration of (sn, s3) designs.
Among other things,
one could get upper and lower bounds for the number of types of
designs.
As far as the practical enumeration is concerned, it is
possible to enumerate with the methods developed here, all the
(sn, sk) designs for n
19
(k "n).
'rne last chapter is concerned with the weight distribution
of a code.
The upper bounds developed here has to be compared with
the real values of the weight distribution for some codes where this
is known.
\-le
have also to compare this bound with some other
bounds and study their respective behavior.
REFERENCES
Assmus, E. F., Gleason, A. M., Mattson, H. F., (1965), SYlvania
~ry Report, No.4, Contract AF (604)-851'6 AFCRL
IfEiOtOrd Mass.
Bose, R. C., (1947), ''Mathematical Theory of the Symmetrical
factorial designs," Sankm, Vol. 8, pp. 107-166.
Bose, R. C., (1960), "On sane connections between the design of
experiments and infomation theory," L'Institut
international d, statistiguI, 32. s.ssion, pp. 1-15.
Buchner, M. M., Jr., (1966), "COlI1puting the speo1:rum of a binary
group code," Bell S:'{stem Technical Journal, Vol. 45,
p. 441.
Burton, R. C. and Connor, W. S., (1957), "On the ideRtity relationship
for fractional factorial designs of the 2 series,"
.
Annals of Hath_tical Statistics, Vol. 28, pp. 762-767.
•
Burton, R. C., (1964), "An application of convex sets to the
construction of error correcting codes and factorial
designs," Ph. D Thesis, 1964, Chapel Hill.
Calabi, L. and MYrVaagn.s, E., (1963), "On the minimal w.ight of
binary group codes," Sgi*ntific Report No.7, Parke
Matheatics Laboratory.
Calabi, L. and MYrVaagnes, E., (1963), "On the weight of the elements
of binary group codes," Scientific r.port, No.5, Parke
Mathematics Laboratory.
Fisher, R. A.., (1942), "The theory of confounding in factorial
experiments in ralation to the theory of groups,"
Annals of lugenics, (11), pp. 344-353.
Forney, G. 01, (1967), "Concaterated codes," R.,.arch monograph
No. 37, M.I.T. Press, Cambridge, Mass.
Goethals, J. M., (1966), "Analysis of weight distribution in binary
cyclic cod.s," 1m Infomat1on Theory, IT-12, p. 401.
Gupta Hansraj, (1958), "Tabl.s of Partitions, It University Press,
Cambridge,
•
101
Kasami. T., (1961S), t'W"eight distributions of BCH codes," Coordinated
Science laboratory. Illinois University.
Kasami. T•• Lin, S., Peterson. W. W., (1966), "Some results on
weight distributions of BCH codes," IEEE Information
Theorv~ IT 12, p. 274.
Kasami. T., Lin, S., Person. ~v. W., (1966), "Some results on cyclic
codes whic~ are invariant under t.J,e affine group,"
Univeristy of Hawaii, Hawaii.
McWilliams, J., (1963), "A theorem on the distribution of weights
in a systematic code." Bell Svst8ll1 Technical Journal.
Vol. 42, p. 79.
Peterson. W. W., (1961), "E:rror correcting codes." M.I. T Press.
Peterson, W. W., (1965), "On the weight structure and symmetry of
BCH codes," Scientific report. No.1, July, 1965, Hawaii.
Singleton, R. C•• (1964), ''l1a.dmum distance Q-nary codes."
Information Theory, IT 10, p. 116.
~
Slepian. D.. (1956), "A class of binary signaling alphabets,"
Boll System Technical Journal. Vol. '5. PP. 203-234.
102
Appendix I
WKlIDHT DISTRIBUTION OF THE DISTANCE 4
WEIGHT
3-ARY HAMMING CODE
DISTRIBUTION
10
24
9
20
8
180
7
240
6
60
5
144
4
60
0
1
---------------------_...
WEIGHT DISTRIBUTION OF THE DISTANCE 4
WEIGHT
4-ARY HAMMING
_~_.~.
__
•....~-
com;
DISTRIBUTION
17
504 648
16
2 856 561
15
7 632 048
14
12 680 640
13
14 861 400
12
12 807 324
11
8 582 688
10
4 488 000
9
1 862 520
8
622 710
7
170 544
6
32 640
5
6 120
4
1020
0
1
•
(
(
(
~
.
loglO of the distribution
'!)O-I
.
•
-
lsi
•
•
•
•
•
•
•
•
•
••
••
•••
•
••
••
-
..
•••• •
••
•
•
1,)1
•
•
,~
•
weight distribution of the
7-ary conic code
•
•
...
•
•
':]
1
eo ...
•
•
•:
!
5
5
5
5
j
10
,, ,
,~
10
as'
\<1
~
"to
.et5
'50
f-I
0
W
104
(
(
(
•
UI
logiO of the
11
•
distribution
,a,
•
•
•
•
•
•
•
•
•
•
•
•
•
"
•
10
•
,
rl
.
•
•
I
•
f
,
S
Weight distribution of the 5-ary Conic code
•
•
••
,
~
•
S
...•
• a•
J
.,
•
10
;
U
•
n.
,
n
•
J~
WEIGHT
•
IS
.6i
,
;
1~
18
,
1'1
&.0
U
It
U
(,4
r.r
u
~
0
\.1\
106
AppEmdix II
CLASS OF (S6,s2) DESIGNS
Type 1
1 0 X X X X]
[ 01XXXX
main
effect
first
order
second
order
•
1
third
order
fourth
order
,
.
fifth
order
•
••
107
Type 2
loxxxol
[ o 1 X X X XJ
main
effect
first
order
second
order
- -.....,---l,l----+--........---+1
(z..,) ~
4
5
third
order
fourth
order
fifth
order
(s-4)
10d
Type 3
lOXXXO)
[ OlXXXO
,
,
,
,
5
1
main
effect
first
order
second third
order order
fifth
order
fourth
order
Type 4
,
lOXXXO]
[ OlXXOX
main
effect
first
order
second third
order order
fourth
order
,
,
fifth
order
(s-3)
Type 5
,
1 0 X X 0 01
[o 1 X X X X
first
order
main
effect
second
order
third
order
fourth
order
,
fifth
order
(8-3)
Type 6
l
1 0 X 0 0 0)
01XXXX
main
effect
first
order
1
second
order
third
order
fourth
order
fifth
order
(s-2)
110
Type 7
lOX X 0 0]
[ 01XXXO
first
order
main
effect
t
second
order
third
order
fourth
order
fifth
order
F F F F 1
2 3 4 S
Type 8
[
main
effect
10XOOX]
X X X 0
o1
first
order
second
order
,
,
lJ ,4)
(a.,4,S)
~
third
order
fourth
order
fifth
order
F2F F F 1
3 4 S
F1F2F4FSF6 1
(8-2)
111
Type 9
:»i.ng td a. l'
-.,
lOX 0 X X)
[ o lOX X X
main
effect
first
order
•
second
order
third
order
fourth
order
fifth
order
(8-2)
Type 10
1 0 0 0 0 0]
[ 01XXXX
•
1
main
effect
fir8t
order
second
order
third
order
fourth
order
(I.. !),4. 5•• )
fifth
order
(8-1)
112
Type 11
[
1
0 X 0 0 01
XXX0
o1
1
main
effect
first
order
second
order
0.,'1,5)
third
order
fourth
order
F F2F F 1
1
4 S
FIF2F3F4FS (s-2)
3
fifth
order
F F F F 1
2 3 4 S
Type 12
,
,
,
3
main
effect
first
order
second
order
F F F 1
134
F F F 1
2 3 4
third
order
fourth
order
fifth
order
113
Type 13
,
1000XXl
[o 1 X X 0 0
2.,S,"
main
effect
first
order
second
order
third
order
fourth
order
fifth
order
(8-1)
'!¥pe 14
r1
Lo
0 X X 0 0]
,
1 X0 X0
(\I~)
main
effect
first
order
(,1,5)
second
order
third
order
fourth
order
F F F 1
1 3 4
F1 F2F4F 1
S
F1F2F3F4FS (8-2)
F2F F 1
3 S
!
fifth
order
114
Type 15
I oI 01 X0 X0 X0 XJ0
main
effect
•
first
order
second
order
third
order
ll,I,"f,5
fourth
order
I
fifth
order
(8-1)
Type 16
10XXXO]
( 010 0 0 0
,
'.l."'.S
main
effect
second
order
first
order
third
order
fourth
order
fifth
order
Type 17
1 0 X X 0 00]
[0 1 X 0 0
main
effect
first
order
second
order
•
~ '.")
third
order
fourth
order
F1F2F3F4 (s-2) ---
,
,
L
!
fifth
order
115
Type 18
lOX 0 X OJ
0 0
ro lOX
main
effect
first
order
second
order
third
order
fourth
order
fifth
order
tIpe 19
l~
main
effect
F2 1
oXX0
1 0 0 0
first
order
~]
,
t',\,""l
fourth
order
second
order
third
order
F F F4 1
1 3
F1F2F F (s-l) --3 4
,
1.
fifth
order
116
Type 20
,
main
effect
first
order
second
order
F F2 I
1
FIF 2 t'3 (s-2)
third
order
fourth
order
fifth
order
FI F3 I
F2F3 I
Type 21
( ,,~)
main
effect
first
order
second
order
fourth
order
fifth
order
117
\
'---
Type 22
[~
oX0 0
100 0
g]
•
"I,~)
main
effect
first
order
second
order
F 1
2
F1F 1
3
F F F (s-l)
1 2 3
t
third
order
fourth
order
fifth
order
third
order
fourth
order
fifth
order
Type 23
f100000]
1 0 0 0 0
Lo
main
effect
first
order
second
order
118
/'
l. / ....
1 0 0 0 X
o 100 X
[ U 0 lOX
U 0 u 1 X
main
effect
-.I
¥
Type 1
first
O.L'der
--.'
XJ
X
X
X
I
second
order
I.
..---+-~----
fourth
order
third
order
2
1
F'J.F 2F:l 4 {6-3)
F "F F 4F l!'6 (s
2 3
5
F F F 1
1 2 4
F J..F F F (s-3)
2 3 5
F1F 3l"4F51"6
F F F 1
1 2 5
F F F F (6-3)
1 2 3 6
F.F"F
1.. ~ 4F, F6
F F F 1
1 2 6
FJ.F2F4F
(6-3)
F J..F 2lt'3F5F b
F F F 1
1 3 4
IJ'lF F F (6-5)
2 4 6
F1F2F3F4F6
F F
1 3
1
F F F F (6-3)
1 2 5 6
F1F2F3F4F5
F F F 1
1 3 6
F1F F4F (6-5)
3 5
F F F 1
1 4 5
l"lF F 4F 6 (6-3)
3
F F F F (6-5)
F F2F
1 3
l',
F F F 1
1 4 6
5
1 3 5 6
F1F F6 1
5
FJ..F 4F F 6 (6-3)
5
F F F 1
2 3 4
Fi'3Fl~F5 (s-3)
F F F 1
2 3 5
F F F F
F~l3F6 1
1"2F 31"5F 6 (6-5)
F 2 F 4 l<'5 1
F ~l4F5F 6 {s-3)
F F F 1
2 4 6
l"3F 4F ':/6 (s- 3)
F F 1"6 1
2 5
F3F'4F5 1
F F F 1
3 4 6
F F Fb 1
35
F4F F6 1
5
2 3 4 6
(s-3)
"'il"th
order
-l.f.s+6)
3 2
(6 -5s +1us-1U)
119
Type 2
lOOOOX]
010 0 X X
[
main
effec c
o
0 lOX X
000 1 X X
second
order
F 1"
.L 6 1 F 1 F 21"3 1
first
order
F~l':)F"
1
~'lF2F5
1
.l ... ...
F.LF3F4 1
~11F3It'5
F
~'
.L
F
4 5
1
1
F2F F4 1
5
F2F F 1
35
F2FLj.~'5 1
F2~13Fb 1
F2F4F6 1
.If' F F 1
2 5 b
F3F.l5 1
F F4Fb 1
3
F F F6 1
35
F'Lj.F 1"6 1
5
third
order
fourth
order
2
FIF2F3F4 (8-3) F"'3F4F5F6 (5 -45+0)
2
F1F2F3F5 (8-3) F1F3FLj.F5FO (s -£+8+5)
F1F2F F6 (8-2) F l'2F 4"5F 6 (s 2 -48+')
3
£:
F F F F (8-3) I'.LF rJ'?l5F6 (s -Lj.s+,)
1 2 4 5
2
F1F2"4F6 (8-2) Fl'2F3F4 F6 (s -4s+5)
F F F F
1 2 5 b
(8-2) ~-'.LF ~'5F 4F5
FIF3F4~'5 (8-3)
F1F F4Fb
3
F1F F Fb
35
F1FLj.F F6
5
F2F F4F
3
5
F2F F4F6
3
F2F F F6
35
F2F4F F6
5
F F4F F6
3 5
(8-2)
(8-en
(s-2)
(8-5)
(8-3)
(8-3)
(8-3)
(8-5)
fifth
order
2
{s -t+s+6)
120
T~'
rooooo]
0100XX
0010XX
.5
000lXX
main
effect
'11
tif'8t
order
second
order
third
order
tourth
order
(8-'1)
r~"4 1
'.J','4'5
'~,'4'5'6
'2','5 1
'~"4'6
' 1','4'5'6
'2','6 1
'1'~4'5'6
'2'4'5 1
'1','5'6
'1'4'5'6
'~4'6 1
'{4'5'6
'1'1','5'6
'1'1','4"6
'~"6 1
'1'~,'4'5
'/4" 1
''/4'6 1
'4'{6 1
""'6 1
fitth
order
(S'.'82+4)
(82.48+6)
(82_1)
121
\
'-----
.
Type 4
-- ~
1 0 0
o1 0
001
[
o0 0
main
effect
0
0
0
1
0 X]
0 X
XX
XX
first
order
F F 1
1 6
second
order
F F F (s-2)
1 2 6
third
order
F2F F F (s-3)
3 4 S
fourth
order
F2F3F4FSF6
2
(s -4s+6)
F F 1
1 2
F F F
1 4 S
1
F F F F6 (s-3)
3 4 S
FIF3F4FSF6
2
(s -4s+6)
F F 1
2 6
F F F
1 3 4
1
F F F F (s-2)
2 3 5 6
FIF2F4F5F6
2
(s -3s+3)
F F F
1 3 5
1
F2F3F4lt'6 (s-2)
FIF2F3F5F6
2
(s -3s+3)
F F F
2 3 S
1
Fl 4FSF6 (s-2)
FIF2F3F4F6
2
(s -3s+3)
F F F
2 3 4
1
F F F F (s-3)
1 3 4 S
FIF2F3F4F5
(s 2-4s+6)
F2F F
4 5
1
F F F F (s-2)
1 2 3 5
F F F
3 4 6
1
F F F F (s-2)
1 2 3 4
F F F
3 5 6
1
F F F F (s-2)
1 2 4 S
F4F F6
S
1
F F F F
1 3 S 6
'--
(~-2)
F F F F (s-2)
1 3 4 6
F F F F (s-2)
1 4 S 6
fifth
order
3
2
(s -5s +9s-9)
122
Type 5
oI0000Xl
1 0 0 X0
o
[ 0 1 0 XX
o0 0 1 XX
main
effect
fir8t
order
8econd
order
third
order
fourth
order
F F 1
2 5
F F F 1
1 2 3
F F F3F (8-3)
1 2 4
2
F2F3F4F5F6 (8 -48+5)
F1F6 1
F1F2F4 1
F1F2F3F (8-2)
5
2
FIF3F4F5F6 (a -48+5)
F1F3F4 1
F1F2F3F6 (8-2)
2
FIF2F4F5F6 (8 -48+4)
F1F3F5 1
F1F2F4F5 (a-2)
2
FIF2F3F5F6 (8 -48+4)
F F4F5 1
1
F1 F2F4F6 (8-2)
2
FIF2F3F4F6 (8 -4s+5)
F F F4 1
2 3
F F2F F (a-I)
1 5 6
2
FIF2F3F4F5 (8 -48+5)
F F F F (8-3)
1 3 4 5
F2F3F6 1
F F F4F6 (8-2)
1 3
F2F4F6 1
F F F F (s-2)
1 3 5 6
F F F 1
3 4 5
F1F4F F (8-2)
5 6
F F F 1
3 4 6
F2F3F4F (s-2)
5
F F F6 1
3 5
F2F3F4F6 (s-3)
F4F5F6 1
F2F3F5F6 (8-2)
F2F4F5F6 (s-2)
F3F4F5F6 (s-3)
fifth
order
3
2
(8 -5s +10s-8)
123
''--
Type 6
100000
o1 0 0 X0
o
[ 0 1 0 XX
000 1 X X
',,-~
main
effect
fir8t
order
8econd
order
third
order
fourth
order
F1 1
F2F 1
S
F2F F 1
3 4
F3F F 1
4 S
F1F F3F4 (8-1)
2
F1F2F3F6 (8-1)
2
F2F3F4FSF6 (s -48+S)
2
F1F3F4FSF6 (s -48+3)
F2F3F6 1
F1F2F4F6 (8-1)
2
F1F2F4FSF6 (8 -2s+1)
F3F F 1
S 6
F1F3F4FS (s-1)
2
F1F2F3FSF6 (s -2s+1)
F2F F6 1
4
F1F F F (s-1)
3 4 6
2
F1F2F3F4F6 (s -4s+3)
!4FSF6 1
F1F F F (s-1)
3 S 6
2
F1F2F3F4FS (s -28+1)
F F F 1
3 4 6
F F F F (8-1)
1 4 S 6
F2F F F (s-2)
3 4 S
F2F3F4F6 (s-3)
F2F F F6 (s-2)
3 S
F2F4FSF6 (8-2)
F3F4F F6 (s-3)
S
fifth
order
3
2
(s -Ss +7s-3)
124
Type 7
1000XX]}
&lngular
010 0 X X
o0 1 0 X0
[000
1 0 X
main
effect
first
order
second
order
third
order
fourth
order
F F 1
1 2
F F F 1
1 3 4
F F F F (a-2)
1 2 3 4
2
F2F3F4F5F6 (s -4a+4)
F3F5 1
F1F F6 1
3
F1F2F3F (a-I)
5
2
F1F3F4F5F6 (a -4a+4)
F4F6 1
F1F4F5 1
F1F2F3F6 (s-2)
2
FIF2F4F5F6 (a -4a+4)
F1F5F6 1
F1F2F4F5 (a-2)
2
FIF2F3F5F6 (a -4s+4)
F2F3F4 1
F1F2F4F (a-I)
6
2
FIF2F3F4F6 (s -4s+4)
F2F F 1
3 6
F1F2F5F (a-2)
6
2
FIF2F3F4F5 (a -4$+4)
F F4F 1
2
5
F1 F F F (a-2)
3 4 5
F2F F 1
5 6
F F F F (a-2)
1 3 4 6
F F F F (8-2)
1 3 5 6
F1F4F5F6 (8-2)
F2F3F4F5 (8-2)
F2F3F4F6 (a-2)
F2F F F6 (8-2)
3 5
F2F4F5F6 (8-2)
F3F4F5F6 (8-1)
fifth
order
3
2
a-58 +108-7
,
125
s/
Type 8
---+-- - .I
1 0 0 0 X OJ
o1 0 0 X0
[ o 0 lOX 0
000 1 X X
main
effect
first
order
second
order
third
order
fourth
order
F F2 1
1
F F F3 (s-2)
1 2
F F2F F (8 2-3s+3)
1
3 S
2
F2F3F4FSF6 (s -3s+3)
F1F 1
3
F1F2FS (.-2)
F1F2F4F6 (s-2)
2
FIF3F4FSF6 (s -3s+3)
F1F S 1
F1F FS (s-2)
3
F2F3 1
F1F4F6
F2FS 1
F2F FS (s-2)
3
F1 F4FSF6 (s-2)
F FS 1
3
F2F4F6
1
F F F4F6 (s-2)
2 3
F3F4F6
1
F2F F F6 (s-2)
4 S
F4F F6
S
1
F3F F F (s-2)
4 S 6
1
2
FIF2F4FSF6 (s -38+3)
F1F3F4F6 (s-2)
2
FIF2F3F4F6 (s -3s+3)
fifth
order
3
2
s -4s +6s-4
126
Type 9
1 0 0
o1 0
o
[ 0 1
000
main
effect
0
0
0
1
X OJ
0 X
X0
XX
fir8t
order
8econd
order
third
order
F F 1
1 3
F F F 1
1 2 4
F F F F (8-2)
1 2 3 4
F F 1
3 S
.F F F 1
1 4 6
F F F F (8-1)
1 2 3 6
2
F2F3Fl sF6 (a -4a+4)
2
FIF3F4FSF6 (a -2a+l)
F F 1
1 S
F F F 1
2 3 4
F F2F F (8-2)
1
4 S
2
FIF2F4FSF6 (a -4a+4)
F2F 1
6
F F F 1
2 4 S
F F2F F (a-2)
4 6
1
2
FIF2F3FSF6 (8 -28+1)
F F F 1
3 4 6
F F F F (8-2)
1 3 4 6
2
FIF2F3F4F6 (a -48+4)
F F F6 1
4 S
F1 F4F F6 (8-2)
S
2
FIF2F3F4FS (a -2a+l)
fourth
order
F2F3F4F (a-2)
S
F2F3F4F (8-2)
6
F F F F (8-1)
2 3 S 6
F F F F (8-2)
2 4 S 6
F F F F (a-2)
3 4 S 6
F F F F (a-I)
1 2 S 6
fifth
order
3
2
a -Sa +78-3
127
Type 10
1
1000XO]
o1 0 0 X0
o
0 1 0 X0
[
000 1 X 0
main
effect
-
first
order
second
order
third
order
fourth
order
F F 1
1 2
F F2F 8-2
1
3
F F F F 8 2-3s+3
2 3 4 S
F1F2F3F4FS
F1F 1
3
F F2F 8-2
1
4
F F F F 82-38+3
1 3 4 S
F1F 1
4
F F2F 8-2
1
S
F1F S 1
F F F 8-2
1 3 4
2
F F F F 8 -38+3
1 2 4 S
2
F F F F 8 -38+3
1 2 3 S
F F 1
2 3
F F F 8-2
1 3 S
2
F F F F 8 -38+3
1 2 3 4
F2F 1
4
F F F 8-2
1 4 S
F2F 1
S
.F F F 8-2
2 3 4
F F 1
3 4
F F3F 8-2
2
S
F F 1
3 S
F F F 8-2
2 4 S
F F 1
4 S
F F4F 8-2
3
S
S
3
2
-48 +68-4
128
1)pe 11
1 0 0
010
[ o0 1
000
XXl
0
0 X0
0 X0
100
main
effect
first
order
second
order
third
order
fourth
order
F4 1
F2F 1
3
F2F F (s-2)
3 S
2
F2F F F (s -3s+2)
3 4 S
2
F1F3F4FSF6 (s -3s+2)
F2F 1
S
F2F3F4 (a-1)
F1F4FSF6 (a-1)
2
F1F2F4FSF6 (s -3s+2)
F F 1
3 S
F2F F (a-1)
4 S
F F3F F (a-2)
S 6
1
2
F1F2F3FSF6 (a -3a+3)
F3F4FS (s-1)
F1F3F4F6 (a-1)
F1F2F3F4F6 (s2_ 3s+2)
F1F2F6
1
F1F2FSF6 (a-2)
F1F3F6
1
F1F2F4F6 (a-1)
F1FSF6
1
F1F2F F (a-2)
3 6
fifth
order
(a 3-48 2+68-3)
•
129
Type 12
is'
o1000XXJ
1 0 0 XX
o
0
100 0
[
000 100
main
effect
first
order
second
order
third
order
fourth
order
F 1
3
F F (8-1)
3 4
F1F F6 1
S
F F F F (8-1)
1 2 3 S
2
F2F3F4FSF6 8 -28+1
F1F2F6 1
F F F F (8-1)
1 2 3 6
2
FIF3F4FSF6 8 -28+1
F1F2FS 1
F F2F F (8-1)
1
4 S
2
FIF2F4FSF6 8 -48+3
F2F F 1
S 6
F F2F F (8-1)
1
4 6
2
FIF2F3FSF6 8 -4s+3
F1 F2F F6 (8-3)
S
2
FIF2F3F4F6 s -2s+1
F F F F (8-1)
1 3 S 6
2
FIF2F3F4FS 8 -2s+1
F 1
4
F F4F F (8-1)
1
S 6
F2F F F (8-1)
3 S 6
F F F F (8-1)
2 4 S 6
fifth
order
3
2
(6 -46 +76-3)
130
Type 13
o1 10 00
[o0 1
000
main
effect
0
0
0
1
r
X 0]
X0
0 X
0 X
&
first
order
third
order
fourth
order
F F 1
1 2
F F2F F (s-l)
1
3 4
2
F2F3F4FSF6 (6 -2s)
F1F 1
S
F1F2F F (s-l)
3 6
2
FIF3F4FSF6 (s -2s)
F2F 1
S
F1F2F F (s-l)
4 6
2
FIF2F4FSF6 (s -2s)
F F 1
3 4
F F F F (8-1)
1 3 4 S
2
FIF2F3FSF6 (s -28)
F F 1
3 6
F F F F (s-l)
1 3 S 6
2
FIF2F3F4F6 (s -2s)
F F 1
4 6
F F4F F (s-l)
1 S 6
2
FIF2F3F4FS (s -2s)
F F3F4FS (s-l)
2
•
F2F3FSF6 (s-l)
F2F4F F (8,..1)
S 6
fifth
order
3 2
(s -58 +48+5)
•
1)1
.
!VRe 14
6
1000XXJ
010 0 0 X
o0 1 0 X0
[
o0 0 1 0 0
main
effect
first
order
second
order
third
order
fourth
order
F 1
4
F F 1
3 S
F F F (s-l)
3 4 S
F F F F (s-l)
1 2 3 4
2
F2F3F4FSF6 (s -2s+1)
F2F6 1
F F F (s-l)
2 4 6
F F F F (s-2)
1 2 3 S
2
FIF3F4FSF6 (s -3s+2)
F F F
1 2 3
1
F1FZF F (s-2)
3 6
2
FIF2F4FSF6 (s -3s+2)
F F2 F
1
S
1
F1F2F4F (s-l)
S
2
FIF2F3FSF6 (s -2s+4)
F F F
1 3 6
1
F1 F2F F (s-2)
S 6
2
FIF2F3F4F6 (s -3s+2)
F F F
1 S 6
1
F F F F (s-l)
1 3 4 6
2
FIF2F3F4FS (s -3s+2)
F F F F (s-2)
1 3 S 6
F F F F (s-l)
1 4 S 6
F2F F F (s-l)
3 S 6
fifth
order
3
2
(s -Ss +6s-3)
1)2
+Eo
,
Type IS
1
[lOOOXO]
o1 0 0 X0
l.
~S
o0
1 0 X0
000 1 0 X
/?a
main
effect
first
order
second
order
third
order
fourth
order
F F 1
4 6
F F F (s-2)
I 2 3
2
F F F F (s -3s+3)
I 2 3 S
2
F2F3F4FSF6 (s -3s+2)
FI F2 1
F F F (s-2)
I 2 S
F1F7. F4F6 (s-l)
2
FIF 3F4F SF6 (s -3s+2)
F F 1
I 3
F F F (8-2)
I 3 S
F F F F (8-1)
I 3 4 6
2
FIF2F4FSF6 (8 -38-\-2)
F F 1.
I S
F F F (8-2)
2 3 S
F F F F (s-l)
I 4 S 6
2
FIF2F3F4F6 (8 -3s+2)
F F 1.
2 3
F F F F (8-1)
2 3 4 6
F F 1
2 S
F F F F (s-l)
2 4 S 6
F F 1.
3 S
F F F F (8-1)
3 4 S 6
fifth
order
(s3 _4s 2+6s- '3)
•
133
.,
'---
Type 16
1000XO]
o1 0 0 X0
o
[ 0 1 0 X0
000 100
main
effect
first
order
second
order
third
order
fourth
order
F 1
4
F F2 1
1
F F2F (8-2)
1
3
F2F F F (8 2-38+2)
3 4 S
FIF2F3F4FS
F1F 1
3
F F2F (8-1)
1 4
F F 1
1 S
F F2F (8-2)
1 S
F2F3 1
F F3F4 (s-l)
1
F F 1
2 S
F F3F (s-2)
1 S
F1F3F4FS (8 2-38+2)
F1F2F FS (8 2-38+2)
4
F1F2F F (8 2-3s+3)
3 S
F F2F F4 (8 2-3s+2)
1
3
F F 1
3 S
F F4F (8-1)
1 S
F2F3F4 (8-1)
F F3F (5-2)
2 S
F2F4F (8-1)
S
F F4FS (8-1)
3
(5 3_45 2+65_ '3)
134
Type 17
s
1000XX]
o1 0 0 X0
001
000
[
o0 0 1 0 0
main
effect
first
order
second
order
third
order
fourth
order
F 1
3
F F 1
2 S
F F F 1
1 2 6
F F F F (s-2)
1 2 S 6
2
F1F3F4FSF6 (a -28+1)
F 1
4
F F 1
3 4
F F F 1
1 S 6
F F F F (8-1)
1 4 S 6
2
FIF2F4FSF6 (s -38+2)
F F F (a-I)
2 3 S
F1F2F F6 (a-I)
4
2
FIF2F3FSF6 (s -38+2)
F F F (a-I)
2 4 S
F F F F (s...l)
1 2 3 6
2
FIF2F3F4F6 (8 -2s+1)
F F F F (s-l)
1 3 S 6
F F F F (8 2-28+1)
2 3 4 5
fifth
order
3
2
8 -4s +58-2
135
Type 18
[lOOOXO]
o1 0 0 0 X
o0
o0
1
1 0 X0
0 100
1.&,,,)
5
3
main
effect
first
order
second
order
third
order
fourth
order
F4 1
F2F6 1
F F3FS (a-2)
1
F1F3F4F (a 2-3&+2)
S
2
F2F3F4FSF6 (a -2a+l)
F1F3 1
F2F4F6 (a-I)
F2F3FSF6 (a-I)
2
FIF2F4FSF6 (a -28+1)
F1FS 1
F3F4FS (s-1)
. F1F2F3F6 (a-I)
2
FIF2F3FSF6 (8 -38+2)
F F 1
3 S
F F F (a-1)
1 4 S
F F2F F (a-I)
1 S 6
2
FIF2F3F4F6 (8 -28+1)
F F F4 (a-1)
1 3
fifth
order
3 2
(a -4a +S8-2)
136
Type 19
1000XX]
010 0 0 0
[o0 1 0 0 0
00010 0
main
effect
first
order
second
order
third
order
fourth
order
F 1
2
F2F3 (s-l)
F1 FSF 1
6
F F F F (s-l)
1 2 S 6
F3 1
F2F4 (a-I)
2
F2F F4 (a -2a+l)
3
F F FSF6 (a-I)
1 3
F4 1
F F4 (s-l)
3
FIF3F4FSFn
(s 2-2a+l)
FIF2F4FSF6
2
(a -2s+1)
FIF2F3FSF6
2
(a -2a+l)
F F4FSF6 (a-I)
1
fifth
order
3
a -38+3s-1
137
Type 20
1 0 0 0 X OJ
010 0 0 X
[ o 0 100 0
000 100
U,S)
main
effect
first
order
second
order
third
order
F 1
3
F F4 (s-l)
3
FI F3FS (s-l)
F4 1
F F
I S
1
F F FS (s-l)
I 4
F F F4F (s 2-2s+1)
I 3 S
2
F2F3F4F (8 -2s+1)
6
F2F6
1
F2F F (a-I)
3 6
F F2F F (8-1)
I
S 6
It,l>)
fourth
order
FI F2F4FSFf)
2
(8 -2a+l)
FIF2F3FSF6
(a 2-28+1)
F2F4F6 (9-1)
fifth
order
s 3-38 2+38-1
\
'-
Type 21
1 0 0 0 X 0]
o1
o0
[
o0
0 0 X0
1 0 0 0
0 100
main
effect
first
order
second
order
third
order
F 1
3
F F (s-l)
3 4
F F2F (a-2)
1
S
2
F2F F F (s -2s+1)
3 4 S
F 1
4
F F
1 2
1
F F F (8-1)
1 2 3
F F
1 S
1
F F2F (s-l)
4
1
1
F F
2 S
F F F (a-I)
1 3 S
F1F4FS (8-1)
F F F (a-I)
2 3 S
F F F (a-I)
2 4 S
2
F F F F (s -2s+1)
1 3 4 S
2
F F F F (s -3a+2)
1 2 4 S
2
F F F F (a -3s+2)
1 2 3 S
2
F1F2F F (8 -28+1)
3 4
fourth
order
FIF2F3F4FS
2
2
(8 -48 +Ss-2)
139
Type 22
1 0 0 0 X OJ
01000 0
[ o 0 100 0
000 100
main
effect
first
order
second
order
third
order
F2 1
F F 1
1 S
F F2F (s-1)
1
S
2
F F F F (s -2s+1)
1 3 4 S
F 1
3
F2F3 (s-1)
F1F F (s-1)
3 S
2
F F F F (s -2s+1)
1 2 4 S
F4 1
F F (s-1)
2 4
F F F (s-1)
1 4 S
2
F F F F (s -2s+1)
2
1
3 S
F F (s-1)
3 4
2
F2F F (s -2s+1)
3 4
fourth
order
F1F2F3F4FS
3
2
(s -3s +3s-1)
140
Type 23
o1
0 0 0 0
1 0 0 0
[o0 1 0 0
00010
0]
0
0
0
main
effect
fir8t
order
8econd
order
F 1
1
F F (8-1)
1 2
F F F (8 2-28+1)
1 2 3
F2 1
F F (8-1)
1 3
2
F F2F4 (8 -28+1)
1
F 1
3
F F (8-1)
1 4
F 1
4
F F (8-1)
2 3
F2F (8-1)
4
F F (8-1)
3 4
2
F F F (8 -28+1)
1 3 4
2
F F F (8 -28+1)
2 3 4
third
order
fourth
order
141
[~o 0~ g1 iX iX iJX
main
effect
first
order
second
order
third
order
,•
•
S•
••
fourth
order
F1F2F:l4 1
F1'2F4'5F6 (s-4)
F1F2F:l5 1
F1F3F4F5F6 (8-4)
F1F2F:l6 1
'2'3'4F5F6 (s-4)
F1F2'4F5 1
F1F2F3F4F5 (s-4)
F1'2F4'6 1
F1'2F3F4F6 (s-4)
'1'2F5F6 1
'1F2F3F5F6 (s-4)
F1'3F4'5 1
'1'3Ft/6
1
'1'3F5F6 1
F1F4F F6 1
5
F2'3F4'5 1
F2'3F4F6 1
F2F F F6 1
35
'2'4F5'6 1
'3F4F5F6 1
2-
•
fifth
order
(s2_ 58+10)
1112
1 0 0 X X 0]
o 1 0 XXX
[o 0 1 XXX
main
effect
first
order
second
order
fourth
order
third
order
F2F4F F6
5
F F4F F6
3 5
F F F F,
1 2 4 b
F1F2F F6
5
F1F F4F6
3
F1F F F6
35
F1F2F F4
1 F2,F F4F5F6 (8-4)
3
1 F1'3F4F5F6 (8-3)
1 F F F F '6 (8-3)
12 45
1 F1'2F3'5'6 (s-4)
1 '1'2F3'4'6 (8-4)
1 F1'2F3F4'5 (8-3)
1
3
F1F2F F 1
3 5
F2F F F6 1
3 5
F2F'3F4F6 1
F2')F 4F5 1
F1F2F F6 1
3
fifth
order
143
1-
~1
1
[lOOXXO]
o 1 0 XX0
.4-
o 0 1 XXX
main
effect
first
order
second
order
•
•
third
order
• (3. ,)
•S
fourth
order
F1F2F 1
5
F1F F4 1
Z
F2F4F 1
5
F1F4F 1
5
F1F2F F6 1
3
F1F F F6 1
35
F1F F4F6 1
3
F2F F4F6 1
3
F2F F F6 1
35
F4F F F6 1
35
F1F2F3F4F6 (8-3)
F2F3F4F5F6 (8-3)
F1F3F5F4F6 (s- 3)
fifth
order
144
~.2
•
•
rOO
o 1 0 xx
X X 0]
0
.~
001 X X 0
main first second
effect order order
I.
•
5
third
order
.1
fourth
order
tifth
order
2
'1'2') 1
'2 F)'4'5 (s-3) '1'2F)'4'5 (8 -4s+6)
'1'2'4 1
F1F)'4'5 (s-)
'1'2'5 1
'1'2'4F5 (s-)
'1')'4 1
'1'2')F5 (s-)
F1'2')'4 (5-)
'1')'5 1
'1'4'5 1
'2')'4 1
'2')F5 1
F2'4'5 1
')'4'5 1
main
effect
first
order
F2F6F4 1
F1F.5F4 1
F1F2F F6 1
5
F1F F F6 1
35
F2F F F6 1
F1F2F3F5F6 (s-4) (;·55+8)
F1F2F F6 1
3
F1F2F F 1
F1F2F4F5F6 (s-2)
3 5
3 5
F F4F F6 1
3 5
F1F F4F6 1
3
F1F2F F4 1
3
F F F F 1
2 3 4 5
F1'3F4F5F6 (s-3)
F2F F4F5F6
3
(s-3)
F1F2F3F4F6 (s-'3)
F1F2F3F4F5 (s-3)
146
1
~2.
rOO
0XJ
o1 0 X
0 X
0
o0
main first
effect order
,
1 XXX
second
order
F F 1 F "4'5 1
2 5
1
4
,
~
fourth
order
third
ol"der
F3'4F5'6 1
F2'3'4"5'6 (s-2)
'2'3'4F6 1
F1'3"4"5'6 ('-3)
'1'3F5F6 1
F1'2'4F5'6 (s-1)
F1F2F:l6 1
F1F F4F 1
3 5
F 2F3'5'6 (s-2)
'1'2"3'4 1
F1'2F3'4F5 (s-2)
l'
F1'2'3'4'6 (8-3)
fifth
order
(82_5s+6)
147
t
•
1 0 0 X X 0]
[ o1 0 X0 0
001 X X X
main
effect
I-
;
first
order
second
order
third
order
'2'lj. 1
F1'2'5 1
F Flj.F F6 1
3 5
F2F F F6 1
35
F1F3Flj.F6 1
F1'lj.F5 1
•
fourth
order
F2F Flj.F F6 (s-2) (s2_lj.s+5)
3 5
F1F Flj.F F6 (s-3)
3 5
F1F2F3F5F6 (s-3)
F1F2F F6 1
F1F2F Flj.F6 (6-2)
3
3
F1F2Flj.F5 (6-2)
'1 F3F5'6 1
\"-.-
fifth
order
148
a.
~~
~1o 01
00 X
XX
0 0]
X
001 0 X X
t
main first second
effect order order
third
order
.--.
3
fourth
order
F F F 1
3 5 6
F F F6 1
2 4
'2F3F4F5 1
F2F3F4F5F6 (s-2)
F1F3'4F6 1
'1 F3'4'5 F6 (8-2)
F1'4F5 1
F1'2F5F6 1
'1'2'4F5'6 (8-2)
F1F2F F6 1
3
F1'2F3F5'6 (s-3)
F F F F 1
1 2 3 5
F1F2'3F4F6 (9-:3>
Ft '2F':l4 1
F1F2F3F4F5 (8-3)
nfth
order
(9 2-5s+7)
149
~2
singular
[1o 00
1 0 1(1
X X ~."
"'-- .
001 0 X X
main
effect
first
order
second
order
third
order
F F2F 1
1 3
F//5F6 1
F1F2' ll5 1
F1'2F4'6 1
-.-
~
~'
'!
fourth
order
F2'3'4'5F6 (s-3)
F1F3F4' 5F6 (5-3)
F1'2F4F5'6 (s-3)
F1F2F5'6 1
F1F F4F 1
3 5
F1F F4F6 1
3
F1F2
F1'3F5'6 1
F1'2F3F4F5 (s-3)
F2F F4F 1
3 5
F2F F4F6 1
3
F2F)F5F6 1
,
t
F1F2F3'5F6 (s-3)
F".l 4F6 (s-3)
fifth
order
(s 2-5s+8)
150
1 0 0 X 0 0]
o 1 0 1. X X
[o 0 1 X X X
Main
effect
first
order
second
order
1
•
third
order
fourth
order
F2F:l5 F6 1
F2F'".l4'5F6 (s-4)
F2F4F5F6 1
F1F)'4F5'6 (s-2)
F2 F')F4F 1
5
'1F2'4'5'6 (5-2)
F2F"/4F6 1
F1F2F3F5F6 (s-4)
F)F 4F5'6 1
F1F2F3F4F6 (5-2)
F1F2F5Ft; 1
F1'2F)F4'5 (5-2)
F1F2 F')F6 1
F1F2 F)F 1
5
F1 F,F F6 1
5
(52_55.....
1S1
(1.,5')
•
~11.
'--
[100XOX]
o lOX X 0
~
•
001 X 0 X
main first second
effect order order
third
order
,•
•
..
.......
J
tourth
order
'1F'J'6 1
F1F2'5'6 1
"2''3'4'5'6 (s-2)
'1'4'6 1
'1'2''3F5 1
F1'2'4'5'6 (8-2)
'1''3'4 1
F2''3'5'6 1
, 1F2'')'SF6 (s- '3)
F'3F4F6 1
F1F'3F4F6 1
F1F2F'3Fl/5 (s-2)
F2'4'S 1
fifth
order
(s2_ 4s+5 )
152
1 0 0 X X X]
o1
000 0
[ 001 X X X
main
first
effect order
second
order
third
order
fourth
order
fifth
order
F1F 4F F6 1
2
F2F3F4F5F6 (8-1) (5 -58+5)
F F F F6 1
1 3 5
F1F3F4F5F6 (8-4)
F1F F4F6 1
3
F1F F4F 1
3 5
F1F2F4F5F6 (8-1)
5
F1F2F3F5F6 (5-1)
F1F2F3F4F6 (5-1)
F1F2F3F4F5 (5-1)
153
r
~1J.
'''--
5
l~
a 1 a0x
x xx 0]a
a a 1 xa a
main
effect
first
order
F)F4
second
order
F1F2F 1
5
F1F2F) 1
F1F)F 1
5
F1F4F 1
5
F2 F)F 1
5
third
order
F2FJF4F5
F1F)F 4F
5
F F F4F
1 2
5
F1F2F3F5
4
•
fourth
order
firth
order
2
(5-2) F1F2FJF4F5 (s -4s+5)
(s-2)
(s-3)
(s-3)
F1F2F)F 4 (s-2)
F2F4F 1
5
1aaXXO]
[aa 1a 1a xa xa ax
m~in
first
effect order
second
order
F F F 1
1 4 5
F1F2F4 1
F1F2F 1
5
F2 Fl l
5 1
•
third
order
('3,£)
fourth
order
firth
order
F1F2F4F (s-) F2F)F4F F6 (s-1) (s 2-4s+)
5
5
F1F)F4F5F6 (s-1)
F1F2F)F F6 (s-1)
5
F1F2F3F4F6 (s-1)
• 1
[1o 00X
X0]
1 0 000
, 4-,
....
001 X X X
main
effect
first
order
second
order
third
order
F1F4F 1
S
•
f
""'
C3 , &J
fourth
order
F1F2F4F (s-1) F1F3F4FSF6 (5-3)
S
F F4F F6 1
F2F;F4F F6 (s-1)
3 S
5
F1F3F4F6 1
F1F2F3F5F6 (s-1)
F1F3F5F6 1
fifth
order
(S2_ 4s+3 )
F1F2F3F4F~ (s-1)
1 0 0 X X OJ
o1 0 X0 0
1001
main
effect
first
order
0 XX
second
order
third
order
fourth
order
F2F4 1 F1F2F5 1
F1F;F 4F6 1 F2F3F4F5F6 (s-1)
F1F4F 1
5
F F F6 1
5 3
F1F2F3F6 1 F1F3F4F5F6 (s-2)
F1F2F4F5(S-2)F1F2F3FSF6 (s-2)
F1F2F3F4F6 (s-2)
fifth
order
(s2_ 45+4)
155
1-
~1Z.
[1a 01 a
0X
0]
xX
0 a
CI,~J
•
aa1x0X
main
effect
first
order
second
order
F2F4 1 F)F 4F6 1
F2F F6 1
3
F1F2F 1
5
F1F4F 1
5
fourth
order
F1F3F5F6 1
F1F3F4F5F6 (s-2)
F1'2 F4F5(s-2) F1F2F)F 5F6 (s-2)
F2F F4F6 (s-2)
3
1.
•
[1o 10 00 XX 00 0]a
\...-
••
o 0 z XXX
first
order
second
order
•
.C3,')
third
order
~18
main
effect
4
•
third
order
fifth
order
2
(s -35+3)
4
*
• (3, 5,')
fourth
order
F F2 1 F1F2F4 (5-2) F1F~l5F6 1 F2F3F4F5F6 (5-2)
1
F2F F F6 1 F1F3F4F5F6 (5-2)
F2FIJ. 1
3 5
F1F4 1
F F4F F6 1 F1F2F)F F6 (5-2)
3 5
5
fifth
order
(5 2-35+3)
156
1 0 0 X 0 OJ
o1 0 0 X0
[ o 0 1 XXX
main
effect
first
order
second
order
third
order
F1F2F)F6 1
F2F)F 4F F6 (s-2)
5
F1F2F4F (s-1) F1F)F 4F F6 (s-2)
5
5
F1F3F5F6 1
F1F2F3F5F6 (s-2)
F2F3F4F6 1
F)F 4F5F6 1
fifth
order
fourth
order
F1F2F3F4F6 (s-2)
2
(s -4s+4)
,
157
x 0]
100X
o 1 0 0 XX
[001XOX
main
effect
first
order
second
order
fourth
order
third
order
F1F2F 1
3
F2F3'4F5 1
'2F3F4F5F6 (s-2)
F3F4F~ 1
F1F F F6 1
3 5
F1F2F4F6 1
~1F3F4F5F6 (s-2)
F 1F 4F 5 1
F2F F6 1
5
fifth
order
(s2_5s+6)
F1F2F4F5F6 (s-2)
F1F2F3F5FS (s-2)
·lif6
F' I F2F
(5-2)
F1F2F3F4F5 (s-2)
T-rpe
Z1
1
•
[1o 1
0 00 0
0 X
XX
'3
001 000
main
effect
F~
,
'----
first
order
second
order
FFF
156
F F2F6
1
F F2F
1 5
F2F F6
5
1
1
1
1
third
order
F F F F6
1 3 5
F2F F F6
3 5
F1F2F F6
5
F1F2F F6
3
F1F2F F
3 5
(s-1)
(s-1)
(s-3)
(s-1)
(s-l)
fourth
order
2
s -4s+3
fifth
order
158
1 0 0 0 0 X]
[
main
effect
first
order
o1
0 0 XX
001 0 X 0
second
order
third
order
F F F 1
1 2 5
F2F.l F6 (s-2)
5
F1F2F 1
3
F1F':{5F6 (s-1)
F F F 1
2 5 6
F1F2 F5F.:) (s-2)
F F F 1
2 3 6
F1F2F·l6 (s-2)
fifth
order
fourth
order
2
(s -4s+4)
F1F2F~"5 (s-2)
(1,4)
~Z1
•
[1o 0000X]
1 0 0 XX
o0
main
effect
first
order
CZ.,5)
1 X0 0
second
order
third
order
fourth
order
fifth
order
F F 1
J 4
F F F 1
1 2 5
F1 F F4'6 (s-1)
3
2
'1F2F3F4F5 (s-1) s -3942)
F F 1
1 6
F F F 1
2 5 6
F F F F (s-2)
1 2 5 6
F2F F4F '6 (s-1)
3 5
159
(1. ~5J
~~
"-
rOOO
0X]X
o 1 0 XX
001000
ml'lin
effect
F 1
3
first
ordE!lr
F1F6 1
second
order
~.3
third
order
fourth
order
F1')F6 ( .. 1) '1'2'4F5 1 F2''],4F5F6 (5-1)
fifth
order
(52..35+2)
F2'4F5F6 1 F1'2F4F5'6 (s-2)
F1F2'3F4F5 (s-1)
~~
[P
(3,1 1 5)
Xl
•
o 1 00 00 00 J
001 X X 0,
'--
main
effect
first
order
second
order
F1F6 1
F']'l.l5 1
F2'3F4'5'6 (5-1)
'2F6 1
'1 F2F6 (s-2)
F1F')'4'5F6 (s-1)
F1F2 1
third
order
fourth
order
F1'2'3F4F5 (s-1)
fifth
order
(52_3~
16 ~1
(1.,+)
1 0 0 0 X X]
o 1 0 X0 X
[ 00100 0
main
effect
first
order
second
order
(I'~
third
order
.1
fourth
order
F1'3F5F6 (8-1) F1'2F4F5'6 (8-2)
f:1rth
order
2
(8 _38+2)
'2 F3F4'6 (5-1) '1F2'3F4'5 (s-1)
F1'2F4F5 1
to
1 0 0 0 0 X]
[o 1 0 X 0 X
001 000
main
effect
/\. .'
I~ClI5)
first
order
second
order
third
order
F1F2 1
F1F6 1
F1F F 1
3 5
F2F3F5 1
F F 1
2 6
F F F6 1
F1F2F F (s-2)
3 5
3 5
F1F2F6 (8-2)
F2F F F6 (5-2)
35
F1F3'5F6 (s-2)
fourth
order
firth
order
161
.\...-
1,,4,','>
~~
[1o
0 0 0 0 0]
1
1 0 XXX
001 000
main
effect
'1 1
first
order
second
order
'1F) (8-1) ---
third
order
fourth
order
'2'4F5'6 1
F1'2F4'5'6 (9-1)
F 1
3
fifth
order
( 82_2s+1)
F2F?l4F5'6 (8-1)
1 0 0 0 X 0]
01000X
[ 001 0 X 0
main
effect
fir~t
order
'1F5 1
second
order
'1')F5 (s-2)
third
order
F2'3'5'6 (8-1)
'1'3 1
F1'2'5'6
'3'5 1
F1F2F')'6 (8-1)
'2'6 1
(8-1)
fourth
order
2
(8 -38+2)
fifth
order
1 0 0
f)
0
n
o1 0 0 X0
[ 001 X 0 0
main
effeot
first
order
seoond
order
•\1,~)
thiro1
order
'1F6 1
'1'2F5F,; (9-1)
F2'5 1
F1F"l4F6 (s... 1)
F')F4 1
F2F'l4F5 (s-1)
~1!
[1o 0000X]
1 0 0 XX
• eL,S)
fourth
orrhlr
fifth
o~er
( ,2. 2.+1 )
~n.'SI
.5
001 000
main
effeot
F) 1
first
order
F1F6 1
seoond
order
third
order
fourth
order
F1'2F5F6 ~ s-2)
(52_3s+2)
F1'2F5 1
F2'3'5F6 (5-1)
F1'2'3'5'6
F2'5F6
F1'2')F5 (5-1)
'1F:l6
(9... 1)
firth
order
163
• (l.S)
main
effect
F3 1
first
order
F1'S 1
second
or;er
.
.
1 0 0 0 X OJ
o1 0 X0 X
[ o 0 1 000
C1.,4,·'J
thirrJ
orner
fourth
orner
~
fifth
order
F1F)F, (s.1) F2')'4'6 (5-1) F1F2'4F5'6 (s-1) (s~2~U
F F F 1
2 4 6
,
•
1.
1 0 0 0 0 X]
o1 0 0 0X
[ 001 0 0 X
main
effect
first
order
F1F6 1
second
order
F1F) 1
F1F2F; (s-2)
F1F2F6 (s-2)
F1F2 1
F1F.l6
F F 1
2 3
F2F'?l6 (s-2)
F2F6 1
F':l6 1
(s-2)
•
.,
third
order
J
•
fourth fifth
order order
1£4
1m! J!t
rOOOOX]
o1 0 0 X0
.,
f.'
1Il.1.n
..
effect
"
n.rR
order
o 0 too
.'1
0
second
ort'er
tb11"d .
oJWJer
tourth
oPder
tltth
order
2
'1'6 1 ~1'3'~ (..1) '1'2','6 ( ..1) '1'2'3','6 (. ~1) ......
t
'2', 1 '?!,'6 (..1)
.b2!
•1
»
,,
eOOOOX]
O1QOOX
001 000
1Ila1n
effect
'3
t1r.t
ol"de.
.eoo~
order
'2'5 1 '1'3'~ (1-1)
'2'3'6 (8.1)
t
S
third
tCNJ'th
order
order
2
'6'1 1 '1'2'6 (1-2) '1'2')'6 ( ...' ....2)
'1'2 1 '1'2', (s-1)
••
--
ftfth
order
---
.'5
XXl
[
main
effect
F 1
3
F2 1
first
order
1 0 00
o 1 000 0
o 0 10 0 0
second
order
F F (s-1) F F F 1
2 3
1 5 6
,j..----.. . . 1.
e"s,')
fifth
order
2
'1'2'5F6 (s-1) F1'2'3F5F6 (5 -28+1)
F1F:l F6 (8-1)
s
1 0 0 0 0 X]
o 1 0 000
[ 00100 0
main
effect
fourth
order
third
order
first
order
second
order
F F6 1
F1F2F6 ls-1}
1
F F (s-1) F F F6 (s-1)
2 3
13
-1.
.~
third
order
fourth
order
fifth
order
166
,
•
1 0 0 0 0 0]
010000
[
001000
main
effect
first
order
F1'2 (s 1)
F1'3 (, 1)
FZ'J (&-1)
:s
\
•
second
order
•
thirn
order
fourth
order
fifth
order