UNIVERSITY OF NORTH CAROLINA
Department of Statistics
Chapel Hill, N, C.
Mathematical Sciences Directorate
Air Force Office of Scientific Research
Washington 25, D. c.
AFOSR Report N9·
1090
SOME TERNARY ERROR CORRECTING CODES AND
FRACTIONALLY REPLICATED DESIGNS
by
R. C. Bose
May, 1961
Contract No. AF 49(638)-213
The connection between the theory of error correcting codes
and the theory of fractionally replicated designs ~s pointed
out by the author. It ~s sho'WIl that the packing problem
plays a fundamental role in both theories. This is the problem of finding the maximum possible number of distinct
points in the finite projective space PG{r-l, p), where
p is a prime or a prime power, so that no d of the points
are dependent. Here we find a configuration of 12 points
in PG(5, 3) so that no 6 are dependent. Ternary error
correcting codes and same fractional factorial designs (with
or without blocking) are deduced.
Qualified requestors may-obtain copies of this report from the ASTIA
Document Service Center, Arlington Hall Station, Arlington 12, Virg:hn1a.. Department of Defense contractors must be established for
ASTIA services, or have their "need-to-know" certified by the cognizant military agency of their project or contract.
Institute of Statistiea
Mimeograph Series No. 295
SOME TERNARY ERROR CORRECTING CODES AND
FRACTIONALLY REPLICATED DESIGNS
By
R. C. Bose
University of North Carolina
1.
Summary
The connection between the theory of error correcting codes and
the theory of fractionally replicated designs was pointed out by the
author in
[)_7.
It was shown that the packing problem plays a funda-
mental role in both theories. This is the problem of finding the maximum possible number of distinct points in the finite projective space
PG(r-l, p), where p is a prime or a prime power, no that no d of the
points are dependent.
Here we find a configuration of 12 points in
PG(5, 3) so that no 6 are dependent.
Ternary error correcting codes and
some fractional factorial designs (with or without blocking) are deduced.
2.
A set of 12 points in PG(5, 3), no five of whicn.aredep~ndBBt..
Consider the 6 x 12 matrix
o
(2.1) C =
[A,I]
=
1
0
0
101122: 0
1
0 000
1 1
0 2 2 1: 0
0
1
0
0 0
1
2
2: 0
0
0
1
0 0
0 1: 0
0
0 010
2 12: 0
0
0
0 1
122 1
1
2 1
0
O~
1 1. 1
1
1 1
001
This research was supported by the United States Air Force
through the Air Force Office of Scientific Research and Development
Command, under Contract No. AF (49) 638-213. Reproduction in whole
or in part is permitted for any purpose of the United States Government.
2
whose elements belong to the Galois field GF(3), and where A is the
matrix formed by the first six columns of
formed by the Jast six columns of
LA, l7.
LA, l7,
Let
0:
1
and I is the matrix
,0:2 "" ,0:6 be the
Let P.J.
column vectors of A, and €1'€2""'€6 the column vectors of I.
and Q.J. be the points in PG(3, 5) with coordinates 0:.J. and e.J.
ly (i
respective-
= 1,2,3,4,5,6). We shall prove
Theorem 1.
If E is the set of 12 points Pi' ~ (i
defined above, then any 5 points of
(a)
~
= 1,2,3,4,5,6)
are independent.
We note that A is a symmetric matrix and that
2
AA'= A
(2.2)
= 2I
The following relations follow at once
A- l = 2A
AO:i
= 2€i
A€i
= O:i
It follows that the transformation
*
plX
= Ax
p ~ 0
,
transformB the point P.J. to Q.J. and vice versa (i
(b)
= 1,2,3,4,5,6).
The points Ql' ~, ~, Q4' Q5' Q6 are independent.
It follows
from (a) that the points P , P , P , P , P , P are independent.
l
2 3
4 5 6
(c) Since Pi has five non-zero coordinates it is clear that Pi
and any four points chosen out of Ql'
dent.
~, ~,
Q4' Q5' Q6 are indepen-
It follows from (a) that Qiand any four points chosen out of
Pl , P2, P , P 4' P , P6 are independent.
3
5
(d) We note that if we choose any three rowed submatrix from A,
any two columns of this submatrix are independent i.e. any 3 x 2 submatrix of A has rank 2.
3
( e ) We shall now show that any three points Qi' Qj' Qk chosen
out of Ql'
~,
... , Q6' together with any two points Pu ' Pv chosen out
of Pl , P2 , ..• , P6 are independent. Let p, q, r be the rows of (A, I)
other than the rows i, j, k. We can suppose without loss of generality
that i < j < k.
Then after suitable row interchanges the coordinates
of Pu , Pv , ~, Qj'
~
are given by the columns of the matrix
a iu
a ju
.a ku
a pu
a qu
(2.3 )
a
a.lV
a.
JV
a
kv
a
pv
a qv
a
rv
ru
1
0
0
0
1
0
0
0
1
0
0
0
0
0
0
0
0
0
It follows from (d) that the rank of this matrix is 5.
Hence
P , P , Q., Q., Q are independent. It now follows from (a) that any
u
v
1
J
""'k
three points Pi' P j , Pk chosen out of Pl , P , •.. , P , together with any
2
6
two points
~, ~
chosen out of
Corollary 1.
~,
Q2"'"
Q6 are independent.
The 12 x 6 matrix
(2.4)
D =
[:J
bas the property (P ), i.e. no five rows of D are dependent.
5
The 11 x 5 matrix D obtained from D by omitting
l
the last row and last column of D has the property (P ), i.e. no four
CorollaX'"'J 2.
4
rows of D are dependent.
l
4
3. Soma ternary error-correcting systematic codes.
Consider a channel which is capable of transmitting anyone
of p distinct symbols.
Such a channel is called a p-nary channel.
We shall assume that p is a prime or a prime power and identify each
symbol with a corresponding element of the field GF(p),
For p
= 3,
t
the channel is called a ternary channel.
Due to the presence of
\noise/a transmitted symbol may be received as one of the other p-l
symbols.
When this happens we say that there is an error in trans-
mitting the symbol.
The symbols successively presented to the
channel of transmission constitute the
ceived constitute the
input
and the symbols re-
output.
Consider n-vectors whose coordinates are elements of GF(p).
The totality, of these vectors forms a vectorspa.ce V • Consider a sublln
space V of V , generated by k independent vectors.
k
n
V consists of
k
k
k
P vectors which may be put in correspondence with a set of p
sages.
mes-
To transmit any message we use the symbols of the corresponding
vector y as
input.
The
is then a vector) * of V .
k
output
r*
(3.1)
then € is the error-vector.
=v
+
If
€
If we define w(€) the weight of €, as
t
the number of non-zero elements in €, then w(e) is the number of errors
•
committed in transmitting
r.
If r
= n-k,
then there exists a vector
space V of rank r, orthogonal to V which consists' of 'all vectors o¥~R~t
r
k
to each vector of V • Let D be an n x r matrix whose column vectors
k
are independent and belong to V •
k
Then the necessary and sufficient
condition for the row vector r to belong to V is
k
5
yD
=0
The matrix D is called the parity check matrix.
*
y D = (y +e)D
=
eD
All error vectors which give' rise to the same
constitute an alias set.
Now
eD
may be . said
Each alias set consists of
vk
~o
error-vectors.
-
If e belongs to an alias set so does e + Yl where Y is any element of
l
Vk ·
The difference of any two error vectors belonBing to the same alias
set is an element of V and thus corresponds to a message.
k
To reconstruct the transmitted vector y from the received
vector y * we must have a rule which enables us to pick a unique e
given e D.
This vector may be called the
corresponding to e D.
leader
of the alias set
Then our decoding rule consists of taking the
transmitted message to be y
= y *-
e.
The set of vectors of V togetk
her with this decoding rule will be called an n-place systematic
p-nary code with k-
inforn~tion
places.
It is clear that a message
will be correctly interpreted if and only if the true error vectors
belongs to set of alias leaders.
When errors occur independently and
with equal probability, errors vectors with smaller weights have
greater chance of occurring.
•
Hence we should follow Slepian's ~9_7
rule of choosing the leader of each alias set to be the vector with
the minimum weight (in case of tie, one of the vectors having the
smallest weight).
Let
E
= (e l ,
e , •.. , en) and let the row vectors of the parity
2
check matrix D be 01' 02]""
on'
Then
6
(3.4)
Thus e:D is a linear function of
,
.5l'~2'
.•. , 5 , the number of nonn
zero coefficients in this linear function being w(e:), the weight of €.
Now suppose that the parity check matrix D has the property
(P2t)' so that no set of 2t vectors from among 51' 5 , .•• ,5 are de2
n
pendent. If in transmitting r , t < t errors have been committed then
0=
wee:)
=to
and there are exactly to non-zero coefficients in e 5 +e 5
1 1 2 2
* *l , e*2 ,· •. ,e*n ) is any other error vector
+... + e 5 in (3.4). If e:=(e
n n
'l(belonging to the same alias set as e: then
5
.• + e *
nn
•. + e 8
n n
",eight *
Now the/wee: ) must exceed to' otherwise
lation between 2t
o
(3.5)
=5
would give a linear re-
or a lesser number of vectors from 5 ,5 , •.• ,5
1 2
n
and this would contradict the fact that D has the property (P2t)' Thus
e: is the leader of the coset corresponding to €D.
There will exist
exactly one linear function of 5 ,o2, ••. ,5 with t or less non-zero
n
1
coefficients equal to r *D, and the coefficients of this linear function
determine € and hence
r= r* -
€.
Thus the code is t error correcting,
i.e. will correct t or a lesser number of errors.
Again suppose that the parity check matrix D has the property
,
(P
+ ) so that no set of 2t+l vectors from among 51 ,o2, ••. ,on are
2t l
dependent. We can show as before that if t 0-< t errOrs are committed,
then the transmitted message r will be correctly interpreted.
now that t+l errors occur in transmitting
r.
Then
r*D cannot
Suppose
vanish.
Also there will exist no linear function of 5 ,5 , •.• ,5 with t < t
1 2
n
0 =
7
non-zero coefficients which will equal I'*D.
together will indicate to us that
there~
These two facts taken
have been at least t+l errors.
However it will be impossible to obtain the exact value of f since
•
there could exist more than one linear function of 51 ,8 2, .. ,,8 with
n
t+l non-zero coeffictents, producing the same value 8
Now consider the special case p
= 3.
= I'*D.
Defining the matrices A
and I as in section 2, we have proved in corollary 1 to Theorem 1,
that
~osaeszes
the property (p,),
Let
C
be given by (2.1).
=!J.,
"1:.7
It follows from (2.2) that
CD
=°
Let V be the vector space generated by the row vectors of C,
6
and let our set of messages correspond to the vectors of V6 • Then we
can take D as the corresponding parity check matrix. We therefore have
the following theorem.
,
Theorem 2.
1'3
= (0,
= (1,
= (1,
1'4
=
1'5
= (1,
1'6
=
1'1
1'2
The 12 place ternary code generated by the vectors
1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0)
0, 1, 1, 2, 2, 0, 1, 0, 0, 0, 0)
1, 0, 2, 2, 1, 0, 0, 1, 0, 0, 0)
(1, 1, 2, 0, 1, 2, 0, 0, 0, 1, 0, 0)
2, 2, 1, 0, 1, 0, 0, 0, 0, 1, 0)
(1, 2, 1, 2, 1, 2, 0, 0, 0, 0, 0, 1)
8
is a two error correcting and 3 error detecting systematic code with 6
information places.
For decoding we calculate Y*D, where Y* is the transmitted vec-
*
If Y D is the null vector then there is
tor and D is given by (2.4).
no error.
* = e.5.
If Y D
~
then
~
is the vector which has e. in the i-th
E
~
If Y*D
place and zero in the other places.
= e.5.+
~
~
e.5. then
J J
€
is
the vector which has e. and e. and in the i-th and j-th positions and
J
~
zero in the other places.
In each case y
= y*-
E.
If Y*D is neither
null nor of the form e.5. or e.5.+ e.5. we conclude that there have
~
~
~
~
J J
been at least three errors.
Again let D be the matrix considered in corollary 2.
l
Let
Yil be the vector obtained by dropping the last coordinate from Yi in
theorem 2, and let C be the 6xll matrix whose row vectors are
l
Then
Let V ,1 be the vector space generated by Yll , 721 , ••. , Y6l and let
6
our set of messages correspond to the vectors of V6 , l'
take D as the corresponding parity check matrix.
l
,
Then we can
Since D has the
l
property (P4) we have the following theorem.
Theorem 3.
The 11 place ternary code generated by the vectors
YII , Y21 , Y3l , Y4l' Y5l' Y6l where ·Yil is obtained from Yi in Theorem
2A, by dropping the last coordinate is a 2 error correcting systematic
code with 6 information places.
For decoding we calculate ylD
*
and proceed as in Theorem 2A.
1
*
where Y is transmitted vector
1
9
Corollary 3.
Each non-null vector of the vector space V6
generated by the vectors Yl , Y2"'"
6 or more.
Y6 in Theorem 2, has weight
If Y is a vector of V then yD
6
from the fact that D has the property
= 0.
The corollary follows
(P5)' since if
y
is of weight
w there is a linear relation among w row vectors of D.
4. Some fractiona.l factorial designs of the type 1k x 3n •
Let P be any prime number.
3
Consider the group ·~'n generated
by the elements
(4.1)
satisfying the relations
~l=~="
2
(4.2)
.=FP=I
n
where I is the unit element of the group.
If G is any element of
the group (in' then by using (4.2), we can express G in the form
a
a
a
1
n
2
G-F
F
(4.3)
- 1 F2
n
where
° -< a.
(4.4)
~
<
p
t
F , F , ••• , F
may be identified with the' n factors of a
2
n
factorial experiment in which each factor can be chosen at anyone
l
•
of the p distinct levels.
at the level Xi (i
(4.5)
The treatment in which the factor Fi occurs
= 0, 1, .•. , n) may be written as
Xl x2
f
1 f2 •
x
f n
n
10
For any gi.v.en G other than the unit element I the totality of
the pn treatments can be partitioned into p disjoint sets wuch that
(4.6)
has the same value (mod p).
The contrasts between these sets carry p-l
degrees of freedom which may be considered to belong to the interaction
G.
All the elements (other than the identity)
p-l
2
G, G , ••• , G
of the sub-group generated by G define the same interaction, since it
is readily seen that any of them generates the same partition of the
set of treatments.
Thus there are (pn_ l)/(p-l) interactions each
carrying p-l degrees of freedom, which account for the pn_ 1 independent
contrasts between the treatments.
Any k independent elements Gl , G , ••• ,Gk of ~n generate a
2
k
to
subgroup !k of order p.
Let
a.
F J.n
n
(4.7)
i
= I,
2, •.• , k
Consider the set S k of treatments
n-
•
for i
•
= ,.
2, .•
~,
k.
We then have p
n-k
treatments in the set.
Hence
they form a l/pk_ th fraction of the totality of all possible treatments .
Let
L be any interaction not belonging to (~k'
nl
(4.8)
L G
l
are called aliases of L.
an alias set.
n2
G
2
\,
~
••• G
k
Then the interactions
O<h.<p-l, i=1,2, .•• ,k
-
J.-
The set of interactions (4.8) is said to be
If in a factorial experiment the responses corresponding
11
to the treatments of the set Skare observed, then we can estiwAte the
n-
sum of all the aliases of L, though L individually cannot be estimated.
Since it is in general more important to estimate lower order
interactions, it is of interest to choose the fundamental subgroup Sfk
in such a way that (for a specified t) no t-factor or lower order inter-
..
action is aliased with another t-factor or lower order interaction, i.e •
any alias set should not contain more than one t-factor or lower order
interaction.
It is readily seen that for this it is necessary and suf-
ficient that every interaction represented by an element of ~k should
have 2t + 1 or more factors.
In particular if each element of C',4-'k has
•.:'J
five or more factors then any main effect or two factor interaction will
not be aliased with any other main effect or two factor interaction.
If we have s further elements Gk+l , Gk+2 , •.• , Gk+s given by
a.
F 1.n
i = k+l, k+2, •.• , k+s
n
'
such that k+s < n, and Gl , G2 ,· •. ,Gk , G +l , ••• , G +s are independent,
k
k
n-k
then the p
treatments of the set S ~ can be subdivided into subsets
n-A.
such that for a given subset
(4.10)
..
,
i
= k+l, .•. ,k+s
where ck+._1, ••• ,c +s are fixed integers less than p. Since each c.1. can
k
be chosen in p different ways the number of subsets is pS, a.nd each subset contains pn-k-s treatments.
If each subset of treatments is assigned
to a different block certain interactions will be confounded
effects.
~h
These interactions are given by
~+s
• Gk+s
(4.11)
O<h.<p-l,
- 1.-
i
= 1,2, ••• ,k+s
block
12
If we want that no t factor or lower order interaction is confounded with any block effect then each interaction in (4.11) must
contain at least t+l or more factors.
•
In particular to leave main
effects and two factor interactions unconfounded each interaction in
(4.11) must be a 3-factor or higher order interaction.
The subgroup
generated by G ,G , •.. ,G + may be called the block subgroup.
l 2
k s
The theory of fractional replication briefly outlined above is
due to Finney ~5_7.
Kishen ~8_7 using !unite geometrical representa-
tion developed earlier by him and the present author ~4_7 generalized
this to the case when p is a prime power.
We shall however not consider
this as we are mainly interested in the case p
= 3.
For the theory of
If in a fractional factorial experiment no main effect or two
factor interaction is confounded with a block effect or aliased with
another main effect or two factor interaction we shall say that main
effects and two factors interaction are measurable(neglecting three
factor and higher order interactions).
Consider in particular the group C}12 generated by Fl ,F2 ,··.,F12
satisfying the relations
F13
= F23
= •••
= F123 = I
It follows from corollary 3, that the subgroup ~6 generated by the
elements G ,G , •.• ,G given below (4.12) is such that each element
l 2
6
contains at least six or more of the letters F (with a non-zero power).
i
13
Gl
=
F
2
G = Fl
2
G = Fl
3
G = Fl
4
G = Fl
5
(4.12)
•
G = Fl
6
F
3
F
3
F
4
F
4
2
F
4
F
2
F
2
2
F
2
2
F
2
2
F
3
2
F
3
F
3
F
5
2
F
5
2
F
5
F
5
F
4
2
F
4
F
5
F6
2
F
6
F
7
F8
F
F
6
2
F
6
F
6
2
F
6
9
FlO
Fll
F
12
We shall now use the above results to obtain some fractional
.
1
n
factorial designs of the type --k x3 where n S 11, The list of de3
signs given is illustrative and not exhaustive,
Suppose there are 11 factors F ,F , ... ,F . Let the funll
l 2
damental subgroup be generated by G ,G , ••• ,G in (4.12) after deleting
6
l 2
the letter F , Since only one letter has been deleted each inter12
(i)
action in the fundamental subgroup will contain five factors or more.
Hence we have
1
3b
x 3
11
fractional experiment in which main effects and
two factor interactions are measurable.
Suppose there are 10 factors F ,F , .•• ,F ' Drop the
lO
l 2
letters F and F
from (4.12). Let the fundamental subgroup be genll
12
(ii)
erated by G ,G , ••• ,G and let the block subgroup have the additional
l 2
5
generator F .
6
,.
Since only one letter viz F has been deleted from the
ll
generators of the fundamental subgroup, each interaction in this subgroup contains five or more letters.
the
bl~ck
Similarly each interaction in
subgroup cmtains four or more letters.
Hence three factor
or lower order interactions are unconfounded with block effects. We
1
10
thus have '"""5 x 3
fractional experiment with the treatments divided
5).
14
into 3 blocks, in which main effects and two factor interactions are
measurable.
If there are 9 factors F ,F , ..• ,F then we can get a
l 2
9
9
,x 3 fractional experiment, with the treatments divided into 9
(iii)
1
?blocks,
by dropping FIO,Fll,F12 from (4.12) and generating the funda-
mental subgroup from G ,G ,G ,G and the block subgroup from G ,G ,G ,
l 2 3
l 2 3 4
G ,G ,G • Main effects and two factor interactions are measurable.
4 5 6
(iv) Suppose there are 8 factors F ,F , •.• ,F . If we drop the
8
l 2
letters F9,FIO,Fll,F12 from (4.12) and generate the fundamental subgroup
from Gl ,G ,G and the block subgroup from Gl,G2,G3'G4,G5,G6 then we
2 3
8
1
get a
x 3 fractional experiment in 27 blocks in which all the main
3
3
effects and two factor interactions are measurable with the exception
of the two factor interactions FIF§,
confounded with block effects.
F2F~, F3F~ and F5F6 which are
If however we generate the block sub-
8
group from G ,G ,G ,G and G then we get a 1 x 3 fractional experl 2 3 4
3
5
3
iment in 9 blocks in which all the main effects and two factor interactions are measurable •
•
t
REFERENCES
Bose, R. C. Mathematical theory of the symmetrical factorial
design, Sankhya, 8 (1947), pp. 107-166.
•
LZ]
Bose, R. C. Mathematics of factorial designs, Proceedings
of the International Congress of Mathematicians, Vol. I
(1950), pp. 54;-548.
Bose, R. C. On some connections between the design of experiments and information theory, Bulletin de l'institut
international statistique, ;8(1961).
Bose, R. C. and Kishen, K., On the problem of confounding in
the general symmetrical factorial design, Sankhya, 5
(1940) pp. 21-;6.
Finney, D. J. The fractional replication of factorial arrangements, Ann. Eugen. Lond., 12 (1945), pp. 291-;01.
Fisher, R. A. The theory of confounding in factorial experiments in relation to the theory of groups, Ann. Eugen.
Lond., 11 (1942), pp. ;41-;5;.
-
L't]
Fisher, R. A. A system of confounding for factors with more
than two alternatives giving completely orthogonal cubes
and higher powers, ~ Eugen. Lond., 12 (1945), pp. 28;-290.
Kishen, K. On fractional replication of the general symmetrical factorial design, J. Ind. Ag. Stat.,l (1948),
pp. 91-106.
- -- - --
,
Slepian, D., A class of binary signalling alphabets,
System Tech. ~, ;5(1956), pp. 20;-2;4.
Bell