titrrVERs:rTt t>1 NO~~H CAROLINA
Department of statistics
Chapel Hill, N. C.
ON A BOUN.D USEFUL IN THE THEORY OF FACTORIAL
DESIGNS AND ERROR CORRECTING CODES
by
R. C. Bose and J. N. Srivastava
April 1963
Grant No. AF-AFOSR-84- 63
Consider a finite projective space PG(r-1,s) of r-l dimensions,
r > 3, based on the Galois field GF, where s = ph, P being a
s
prime. A set of distinct points in ro(r-l,s) is said to be a
non-collinear set, if no three are collinear. The m:l.ximum number of points in such a non-collinear set is denoted by ~(r,s).
It is the object of this paper to find a new upper bound for
~ (r, s ) • This bound is of importance in the theory of factorial
designs and error correcting codes. The exact value of ~(r,s)
is known when either r ~ 4 or when s = 2. When r ~ 5, s > 3,
the best values for the upper bound on II}(r,s) are due to Tal1ini
,[J.P.7 and Barlott! [)}.
"When
s
is even.
Institute of Statistics
Mimeo Series No. 359
Our bound improves these when s
=3
or
ON A BOmID USEFUL IN THE THEORY OF FACTORIAL
DESIGNS AND ERROR CORRECTING CODES
I
by
R. C. Bose andJ. N. Srivastava
University of North Carolina
= ====== ===== = = == ===== == = = = = = = = = = =- - = = == = ==
O.
Summary
Consider a finite projective space PG(r-l,s) of r-l dimensions, r
~
3, based
h
on the Galois field GFs ' where s = l' , l' being a prime. A set of distinct points
in PG(r-l,s) is said to be a non-collinear set, if no three are collinear. 'Fhe'man~ (r, s)
mum number of points in such a non-collinear set is denoted by
object of this paper to find a new upper bound for
~(r,s).
• It is the
This bound is of im-
portance in the theory of factorial designs and error correcting codes.
~(r,s)
value of
is known when either r < 4 or when
s
= 2. When r
the best values for the upper bound on Il)(r,s) are due to Tallini
Barlotti
1.
£];7.
Our bound improves these when
s
=3
or when
s
The exact
~
£12.7
5, s > 3,
and
is even.
Introduction
R. A. Fisher
£!!.7, £17
showed that the maximum number of factors, which can
be accamodated in a symmetrical factorial design in which each factor is at
levels and the blocks are of size
sr
(Where
any main effect or two factor interaction is
Bose
f~7,
s
s
is prime), without confounding
(sr.l)/(s_l).
generalizing Fisherts result proved the following:
Let mt(r,s)
denote the maximum number of points which can be chosen in the finite projective
"
~his research was supported in part by the Mathematics Division of the Air Force
Office of Scientific Research under Grant No. AF-AFOSR-84-63.
2
~
space pG(r-l,s), where
s
is a prime or the power of a prime, so that no
the points are dependent.
The
t
m (r, s) is the maximum number of factors which we
t
can accomodate in a symmetrical factorial design in which each factor is at
levels and the blocks are of size
action is confounded.
t
= 2,
of
r
s , so that no t factor or lower
s
order inter-
Fishers result follows at once by noting that for the case
mt(r,s) is simply the number of distinct points in PG(r-l,s).
1
n
For a fractionally replicated design ~ x s , consisting of a single block
r
s
with s
plots or experimental units, n z:: r + k, a slight modification of Bose's
argument shows that if it is required that no d-factor or lower order interaction,
should be aliassed with a d-factor or a lower order interaction, then the maximum
possible value of n
is
m (r,s).
2d
On the other hand if it is required that no
d-factor or lower order interaction should be aliassed 'With a (d+l)-factor or a
lower order interaction, then the maximum value of
n
would be m + (r, s) •
2d l
The number mt(r,s) also turns out to be important in information theory.
t here is an
s-ary channel, 1.e. a channel capable of transmitting
symbols, then for an
redundancy
r
=n
(n,k) group code, with k
- k, the maximum value of
corrected 'With certainty is
which
d
~d(r,s).
n
m2d+ (r, s ) •
l
distinct
information symbols and fixed
for which
d
errors can be
Similarly the maximum value of
errors can be corrected 'With certainty and
detected is
s
If
d + 1
n
for
errors can be
This parallelism between the theory of fractional repli-
cations and error correcting codes has been brought out by Bose
1"L7.
Thus the problem of finding the maximum value of m (r, s ), and of obtaining
t
by a constructive method the corresponding points of PG(r-l,s) is of some importance.
This problem may be called the packing problem.
this problem are at present know.
bound on mt(r,s) is desirable.
Only partial solutions to
In the absence of a complete solution a good
In this paper we shall consider the case
t
= 3.
For this case the packing problem reduces to finding the maximum number of points
in PG(r-l,s) so that no three are collinear.
collinear set, and ~(r,s) is
Such a set may be called a
~
then the maximum number of points in a non-collinear
set in PG(r-l,s).
Bose
£E.7
showed that for the case
r =
3, (i.e. for a finite projective plane)
(1.1)
~(3,s) = s+l
when
s
is odd,
(1.2)
~(r,s) = s+2
when
s
is even.
For the case
r> 3, s = 2, Bose£E.7 showed that
and the same result was obtained independently by Hannning
£§l,
in connection with
binary group codes correcting one error and detecting two errors.
For
r=4, Bose
LE.7
showed that when
~(4,s)
(1.4)
s
= s 2+1,
is odd
s > 2
and the same result was proved to hold true for the case when
by Qvist £17, the particular case
When
s=4
s
is even (s > 2)
having been obtained earlier by Seiden £27.
r ~ 5, s > 2, the exact value of ~(r,s) is not known.
bounds currently known are due to Tallin1, and Barlotti.
The best upper
Thus Tallini £12,7 has
shown that
~ ( r,s )
< s r-2+
s > 2, r ~ 4
1 ,
the result holding both for odd and even
s.
When
s > 3, Barlotti
£J},
proved the bound given above.· He shows that:
(1.6)
II)(r,s) ~ sr-
2
- (s-5)
r-5
E
i=o
.
s~ + 1,
r
~
5, s
~
7 and odd,
has im..
4
s
(1.8)
In; ( r, s .)
~
s
r-2
r-6
- 26
i
E
S
=5
s = 5, r
1,
-
,
>6 ,
i=o
(1.9)
m;(5,s)
~ s3 ,
(1.10)
m; ( r,s )
~
s
seven
r-2
- s
r-6
E
i=o
s
i
,
seven
,
r
>6 •
We obtain in Theorem 2, section 4, a new bound which is an improvement over these
results when
s = 3, or
same as Bar10tti f s.
s
is even (6 > 2).
For
s = 5, r
=5
In other cases Bar10tti f s bound is better.
our bound is the
For lower bounds on
~(r,s) and other important results on non-collinear sets reference may be made to
Segre
2.
£§l.
Symmetric representation for
t = 3
A set of points in finite projective space PG(r-1,s) of r-l dimensions,
based on the Galois field GF
s
collinear set if no
where
s = ph (p being a prime), is said to be non-
three of the points are collinear.
The set is said to be can-
plete if we cannot add any new point to the set, so that it still retains the
property of non-co1linearity.
The maximum number of points in a non-collinear set
in pG(r-l,s) may be denoted by m;(r,s).
Barlotti
[)7
-
would be denoted by M 1 •
r- ,s
The same number in the notation used by
Let
(2.1)
be a complete non-collinear set in PG(r-l,s).
Nr
= (sr -
Since there are
l)/(S-l) ,
distinct points in pG(r-l,s), there are n
= Nr
- m points
5
n
not contained in the set S given by (2.1).
= Nr
.. m
We will denote bY'
S the set of
points (2.;).
From the property of non..collinearity no line in pG(r-l,s) can intersect the
set S
in more than two points.
intersects
S
it intersects
if
S in a single point, and will be said to be a non-intersector if it
S.
B
Through each of the points
If not
S, if it
in two distinct points , it Will be said to be a tangent to S
contains no point of
of S.
A line 'Will be said to be a secant of
i
in (2.;) there must pass at least one secant
~,A2'
••• Am' B would be a non-collinear set contradicting the
i
property of maximality. Let u
be the number of secants through
i
= 1,2, ... ,n
r .. m), 1 ~ ui ~ lm/~7, where 1!7 denotes the Jargest integer
not exceeding £!7. On every secant there must occur exactly s.. l of the points
B (i
1
= N
B, since each line of
m(r-l,s) has exactly
6+1 points.
The secants can be exhibited in a tabular form as follows, where the i-th row
shows the secants passing through B •
i
Bl ~ll Al12 ,
A ,
B A
2 21l 2l2
(2.4)
•• •
Bl ~2l A1221
A
ijk
... ,
, ... ,
•••
A
A , ... ,
i21 i22
..
.
, ... ,
A
A , ... ,
n2l n22
B2 ~2l A222 ,
,
B
i
B
n
Here the points
.... ,
•••
6
belong to the non-collinear set S
= ~,
Bn belong to the complementary set
s.
The coordinates of B
i
A , ... , Am'
2
and the points
B , B2 , ... ,
l
can be represented by the vector
i = 1,2, ••• , n
=N
l'
- m
and the coordinates of At by the vector
(2.6)
t = 1,2, ••• ,m
The relations between these vectors then can be exhibited as
where
A.'S
and
pIS
are non-zero elements of GF
.
and the
a's
s -
set of vectors ~l' ~2' ... ~m
belong to the
representing the points of S, (i = 1,,2, ... ,n).
The scheme (2p4) or its vector equivalent (2.7) may be ca.lled the symmetric
= 3.
It specifies the structure of the complete non-collinear
••• , Am.
We shall use this scheme for obtaining an upper bound on
representation for t
set
S =A ,
l
~(r,s).
~,
This scheme can also be used for constructing non-collinear sets, but
this will be considered in a separate paper.
3. Proper a.nd improper non-collinear sets
Consider the complete non-collinear set
cannot contain more than
points when
s
S given by (2.1).
s+2 points of S when
s
is odd, from the result due to Bose
duction, since the points of S
contained in
A plane
is even, and more than
£2_7
s+l
mentioned in the intro-
must themselves form a non-
collinear set.
We shall call the non-collinear set
S, improper if there is at least one plane
7
which contains
s+2
points of S.
call the non-collinear set
points of S.
s
be a plane which contains
N 3
r-
3-s:Paces of PG(r-l,s).
s2_ S _ l
We shall
which contains s+2
S = AI' A , ••• , Am
be improper, and let
2
s+2 points of S, which may without loss of
Al'~'
... AS+ 2 •
= (sr-3
• l)/(s-l)
Through the plane
there pass
,
NoW from the result of Qvist
duction, no 3-space can contain more than
contains
must be even.
may be odd or even.
Let the complete non-collinear set
generality be taken to be
s
proper if there is no plane of
S
In this case
In this case
117 mentioned
2
s +1 points of S.
in the intro-
Hence each 3-space
or less points other than AI' A , ... , AS+ • We therefore have
2
2
the theorem:
Theorem 1.
The number of points
m in an improper non-collinear set in
PG(r-l,s), s > 2, satisfies the inequality
m~ (s+2) + (s2 - s - 1)Nr _
3
4. Some useful lemmas.
The number
weight of B••
J.
u
i
of the secants of S, passing through B
i
may be called the
We may write
(4.1)
Each of the points
tribute a weight
of all the points
A, contained in a secant through B.
J.
1/2 to
B..
J.
The weight of B.
J.
may be supposed to con-
is then the sum of the weights
.
A, lying on secants through B , i. e. the weight of all the points
i
A occurring in the i-th row of the scheme (2.4).
8
n
~
Lemma 1.
i=l
1
= "2 m(m-l)(s-l),
ui
The number of distinct secants of the set
secant must contain exactly two points of
the points
Ai and A •
points
and A
Ai
There are exactly
j
j
B.
is exactly m(m-I)/2
since any
Consider the secant passing through
s-l of the points
each contribute a weight
and to no other points
m(m-1)(s-I)/2.
S.
S
B, on AiAj •
1/2 to each of these
Hence the total weight of the points
The
6-1 points,
B is
This proves the Lemma.
We sr...al1 define the weight of a secant AiA as the sum of weights of all the
j
5 .. ].
points
B, contained in AiA , and denote this weight by W(A~j)
Lemma 2.
j
~
i<j
=
W(AiAj )
Consider any point B •
i
n
2
u. ,
i=l
1
~
There are
=
i,j
U
1,2, ••• , m •
secants passing through it.
i
counting the 'Wei_ght of' all the secants, the weight of
B
i
is counted u
Hence in
i
times.
This proves the result.
Lemma. ;.
S = AI' A , ••• , Am
If the non-collinear set
2
is proper
(4.2)
tion
W(Ai Aj ) ~
(4.;)
s-l
~
k=l
be a point of 5
W{Bijk ) ·
Lst
At
distinct from Ai
1/2
to Bijk , there exists a point
~
.
B, '1' B, '2' ••• , B. j
1J
1J
1
If At
,s-l'
Al' A2 , ... As _1 (distinct from Ai' Aj , At)
Hence this plane has
If
~
contributes a. weight
of S (distinct from Ai' Aj , At) such
that the line At~ passes through B
•
ijk
each of the points
and A •
j
contributes a weight
1/2 to
then there exist distinct points
of 5, lying in the plane ~AjAt •
s+2 points of S, which contradicts the fact that
S is
9
can contribute a. weight 1/2 at most to utmost s-2
proper.
Henee At
points
B on AiAj •
Hence the total weight eontributed by the m-2 points ot S
to points of !
not lying on AtAj
on
AiAj
does not exceed (m..a)(s-2}/2.
the other band each ot the points Ai and A
j
of the points Bljl' B1j2" • H, Blj,S_l'
s-l
(4.4)
w(Bijk )
1:
ot the
k=l
This. proves the Lemma.
On
contributes a weight 1/2 to each
Hence
~(s-l) + ~ (m-2)(s-2) •
5. Upper bound on the number of R0ints contained in a non-collinear set in
ro(r-l,s), where
6
>
'2 •
First let the non-collinear set
s .. ~,~, ... ,
Am be proper and canplete.
Then from LeJmnas 2, and ;,
m(m-l)
2
Also
~om
Lemma. 1
>
n -= Nr - m
222
mlm-l) (sw1)
(Nr .. m)
•
Hence from (5.1) and (5.2) we have
m2( s 2- s .. 1) .. m ( s 2 - 26 .. 1) + Nr( 5-2)
-2N
<
r-
0
•
Hence m cannot exceed the positive root or the quadratic equation
'2
2
x (s .. s - 1) .. x
(8 2 .. 25 .. 1) + N (s.. 2)
r
.. 2 N
r
=0
•
i'
..
10
If the non-collinear set S
n > 2.
= 2n ,
where
4.
in the left hand side of (5.4) by (s+2) + (6 2 - 6 - l)N ~
.
is negative
r-/
Hence the upper bound on m given by (3.1) is smaller than the upper
bound on m given by (5.5).
s > 2, r
6
One can check after some calculation that in this case the result of re-
placing x
if r >
is improper and s > 2, then
~
The later bound is therefore valid in all cases where
4. If we have a non-collinear set which is incomplete then we can add
more points to it to make it a complete non-collinear set..
Hence we have the
theorem:
Theorem 2.
If Il)(r,s) denotes the maximum possible number of points in any
non-collinear set in ro(r-l,s), then Il)(r,s) cannot exceed the positive root of
the equation (5.4), if
Hence if
s > 2, r ~ 4.
s > 2, r ~ 4, Il)(r,s):5 ,(r,s) where
2
vCr,s) <
-
__2
2
\!fer,s) i8 given by
3
2
N (8-2)+(S -2s-1)+~N~(S-2) +2N (6 -5-2)+(5 -2s-1)
r
r
2
r
2(S -s-l)
2 1/2
-
7
N = (sr_l)/(s_l) ..
where
r
6. Discussion of special cases, and comparison with previously known bounds.
(a)
Consider the special case
side of (5.4).
s = 3, r ~ 50.
Let
f(x) denote the left hand
Then
where Nr
Let
u
= (2
+ Nr
)/5 •
Then
= 7 .. Nr < 0 ,
2) = 24
>0 •
feu + 1)
feu +
= {3r
- 1)/2
•
, ,.
u
Hence ,}" ( r, s ) lies between u + 1 and u + 2.
~(r,s) ~
N + 12
r
,(r,s) <
..
5
The bost previously known bound due to Tallini
£IE7
:Ls given by €1.5) whieh
in this case reduces to
Hence for this case the bound given by (5.') is always better than the bound
given by (1.5).
(5..4)
s > 3, r .. 5.
Consider the special case
(b)
Denoting the left hand side of
f'(x) we have
by
t{x) •
Let
xL:x(s
2
- s .. 1) .. (s
'2
.. 2s - 1) -
N,(s-217 ..
2N
5
•
'2
(8 .. 26 .. 1) .. N,(e-'2)
u :;;:
2
(8 -a-I)
•
For any c
feu +
e) = (u
:=
Taking
+
c) e (s 2 .. s .. 1) .. 2M,
N fC ( a-2) .. 2_7 + e ( s
5
2 2
.. 28 .. 1) + C (s - s - 1) •
c:= 3(S+1)/(s2 - s _ 1), we see that
> 0
f(s3)
Taking
'2
c
= -1
•
2
117,
f(s3 - 1)
< 0 •
+ [)(S+1)/(8 .. s ..
we find that
,
.
tt
12
Hence V(rIS) lies between s3-1 and 53.
We therefore have
This equals the Barlotti's bound given by the equation (1.1) for the case
s = 5, and improves Barlotti's bound given by the equation (1.9) when
(c)
Consider the case when s
is even,
s > 2 1 r > 6.
8
is even.
.AB before let
f(x)
denote the left hand side of (5.4) and let
U
=
(s2 _ 28 - 1) - N (s - 2)
r
2
•
(s - s - 1)
Then
feu)
f(U+l)
Hence v(rls)
= -2Nr <
0
I
= (s-4)N + (s-2)(2s+l) •
r
lies between U and u + 1. Thus
It is easy to check that in this case u+l is less than the right hand side
of the equation (1.10),
Barlotti.
80
that our bound is an improvement over that given by
,
..
13
REFERENCES
£'l::.7
A. Barlotti" "Una limitazione superiore per il numero di punti apparlenenti a
(k"o) di uno spazio lineare finito~ If Boll. Un. Mat. Ital" 12
una calotta
(1957)" pp. 67-70.
£:?:.7 R.. C. Bose, "Mathematical theory of the symmetrical factorial design" It
Sankhya, 8 (1947), pp. 107-166.
£~7 R. C. Bose, "On some connections between the design of experiments and in-
fo:r.mation theory" If Bulletin de 1 1 Institute International de Statistique,
e
38" 4 (1961) pp. 257-271.
£!!.7 R. A. Fisher, "The theory of confounding in factorial experiments in relation
to the theory of groups" It Ann. Eugen. Lend., 11 (1942), pp. 341-353.
£27 R. A. Fisher, "A system of confounding for factors with more than two alternatives giVing completely orthogonal cubes and higher powers," Ann. Eugen.
Lend., 12 (1945)" pp. 283-290.
£§J
R. W. Hamning, "Error detecting and error correcting codes, ff Bell System
Tech. J., 29 (1950)" pp. 147-160.
£17 B. Qvist, "Some remarks concerning curves of the second degree in a finite
plane, tI Ann. Acad. Sci. Fenn. sere A" I, no. 134 (1952).
£§J
B. Segre, "Le geometrie di Galois," Ann.. Mat., 48 (1957)" pp. 1-97.
£2.7 E. Seiden,
A theorem in finite projective geometry and an application to
If
statistics," proc. Amer. Math. Soc., 1 (1950), pp. 282-286.
£12.7 G. Ta11ini" "sulla k-ca1otta di uno spazio 1inea.re finito, II Ann. Mat., 42
(1956), pp. 119-164.