ORTHOGONAL A R R A Y S OF STRENGTH FIVE

Bodh Raj Gulati

Southern Connect icut State College

O. Summary.

Fractional factorial designs o f strength t = 2u + 1 permit estima-

tion of all the main effects and u- factors or lower order interactions

without aliasing by any (u + 1) factor or lower order interactions.

This paper produces two level orthogonal arrays of strenght five

based on 32, 64, 96 and 128 assemblies. The maximum number

of constraints is established in each case and arrays assuming these va-

lues are effectively constructed. The concept of maximal (k, t) - sets

based on finite projective geometry is exploited in the derivation of

some of these arrays.

1. Introduction.

A k X N matrix A, with entries from a set S of s / > 2, is called an

orthogonal array of stregth t, size N, k constraints, and s symbols if

each t X N submatrix of A contains all possible t X 1 colum vectors

with elements from S with the same frequency X. This array may be-

denoted symbolically by (N, k, s, t ) and X is called the weight or the

index of the array. Then it follows easily that N = Xs t.

The set S may be regarded as the set of s symbols emanating from

Galois field GF(s) , where s is prime or power o f a prime. For example,

the orthogonal array (16, 8, 2, 3) with weight two is given below. It

can be verified that in any 3 X 16 submatrix, e a c h o f t h e eight column

vectors (0, 0, 0), (0, 0, 1), (0, 1, 0), (1, 0, 0), (1, 0, 1), (1, 1, 0), (0, 1, 1),

(1, 1, 1) appears twice in the following matrix.

51

0 1 0 0 0 1 1 1 0 0 0 0 1 1 1 1

0 0 1 0 0 1 0 0 1 1 0 1 0 1 1 1

0 0 0 1 0 0 1 0 1 0 1 1 1 0 1 1

0 0 0 0 1 0 0 1 0 1 1 1 1 1 0 1

0 0 1 1 1 1 1 1 0 0 0 1 0 0 0 1

0 1 0 1 1 1 0 0 1 1 0 0 1 0 0 1

0 1 1 0 1 0 1 0 1 0 1 0 0 1 0 1

0 1 1 1 0 0 0 1 0 1 1 0 0 0 1 1

If k • N matrix A is of strength t, so is any subarray of k' rows

if t ~< k' ~< k. Hence nonexistence of (N, k', s, t) automatically implies

the nonexistence of (N, k, s, t) if k > k'. Again, it may be observed

that if a k • N matrix A is of strength t, it i s a : ;o of strength t' for

all t' ~< t. Hotelling [ 11 ] considered orthogonal arrays of strength two and

two levels with a view to applying factorial designs to chemistry. The

opt imum multifactorial designs considered by Plackett and Burman

[12] are essentially orthogonal arrays of strength two. They have

shown that the maximum number of constraints k, for an orthogonal

array of size Xs 2, s levels and strength two, satisfies the inequality

where [x] is the largest possible integer not exceeding x. For unit

weight, the above inequality implies the well known fact that the maxi-

mum number of orthogonal s • s Latin squares is bounded by s - 1.

The following generalization of the above inequality is due to

Rao [ 131.

Theorem 1.1. (Rao): For an orthogonal array (N, k, s, t), t >~ 2,

the number of constraints k satisfies the following inequalities

(1.2) N - l > ~ C ~ ( s - 1 ) + C ~ ( s - 1) 2 + .... +Ck=(s - 1) u i f t = 2 u

(1.3) N - 1 >~C~(s-1)+C~2(s-1)2+ ... + C~(s-1)u+C~ - I(S--1)u+I

if t = 2 u + 1

52

For t = 2, this reduces to the inequality (1.1). When t = 3, the

Rao's theorem yields the following statement.

For an orthogonal array (N, k, s, t) of strength 3, the number

of constraints k satisfies the inequality

1 (1.4) k ~ < ] + 1.

Bose and Bush [2] considered arrays of strength two and three

and have given an alternative p roo f of the inequalities (1.1) and (1.4).

In a separate paper K. A. Bush [14] has shown that for or thogonal

arrays (s t, k, s, t) of unit weight and arbitrary strength t, the number

of constraints k satisfies the inequality

(1.5) k < s + t - 1 when s is even

(1.6) k ~< s + t - 2 when s is odd prime. An alternative proof leading to the above inequalities using the concept

of subspaces is given in [9,10].

Selden and Zemach [15] have constructed orthogonal arrays of

strength four for different weights. They have analyzed the structure

of the arrays (X2 t, t + 1, 2, t) and have developed a method for exten-

ding an array for X = q2 ~(q odd) to t + n + 1 constraints. However ,

they state that their method is not exhaustive and further examination

of other possibilities is required. We wish to remark that for q -- 1, the

number of constraints obtained by methods of projective geometry is

larger than the number assured by their techniques but for o ther odd

values of q >~ 3, their method is helpful.

Recently Blum, Schatz and Seiden proved the following theorem.

Theorem 1.2 [3]: IfX is odd and X ~< t - 1, then maximum number

of constraints k is exactly t + 1.

A necessary condition for an array to be orthogonal is [2] that for

each nonnegative integer h not exceeding the strength t, k must satisfy

the following set of t + 1 linear equations. ] : k

(1.7) ~_~ n i l = X s t - - 1

1 = 0

53

j~ k"

(1.8 =c cxs ' - h - l <.h<. t j = l

where C~ = 0 for j < h and n O. d e n o t e s the n u m b e r o f co lumns ( o t h e r

than i t h ) tha t have j co inc idences with ita co lu m n of an o r t n o g o n a l

array, i = 1, 2, 3 , . . . , N ; j = 1 . . . . , k .

1.20rthogonal Arrays and Some Related Designs.

F o r a symmet r i ca l factor ia l e x p e r i m e n t with k fac tors each at s

levels, N c o lumns o f an array may be iden t i f ied wi th the t r ea tmen t com-

b ina t ions or assemblies, k rows co r r e sp o n d to the k factors involved

in the e x p e r i m e n t while an ent ry s tands for the level of a f ac to r against

which it is shown. These N assemblies cons t i t u t e a subset o f s k possible

t r e a tme n t combina t ions that we need in a comple t e factor ia l expe-

r iment .

In a comple t e factor ia l design, all main e f fec t s and in te rac t ions up

to o rde r k - 1 are es t imable bu t these take up all the degrees o f fre-

e dom leaving none for the error. To resolve such si tuations, one m ay

ei ther use est imates o f e r ror variance based on previous studies or one

may derive a valid es t imate o f e r ror variance assuming cer ta in higher

order in te rac t ions to be absent . Some t imes , it is no t possible to set up

even a single repl ica t ion because o f the large n u m b e r o f assemblies in-

volved. To o v e r c o m e this d i f f icu l ty , F i n n e y in t roduced f rac t iona l ly

repl ica ted designs using on ly a subset o f the s k t r e a t m e n t co m b in a t i o n s

which would provide est imates o f main e f fec ts a n d l o w e r order in terac-

t ion on the assumpt ion that higher o rde r in te rac t ions are o f no specif ic

interest and thei r con t r ibu t ion in the e x p e r i m e n t is negligible. I f a frac-

t ion 1 / s a of a s k factor ia l sys tem is used, there are obvious ly s k - a

t r e a tme n t combina t ions actual ly tes ted with ( s k - a - 1) degrees o f

f r e e d o m among them. Of the to ta l s k - 1 ef fec ts and in te rac t ions in the

full mode l , s a - 1 are mutua l ly c o n f o u n d e d wi th the total while the

remaining s k - s a will be c o n f o u n d e d is groups o f J , there being

(s k - a _ 1) such groups. In o t h e r words , s k t r e a t m e n t co m b in a t i o n s

are split in to s a dist inct blocks, each o f size s k - a , so that each b lock

is r ep re sen ted by a k X s k - a ma t r ix wi th entr ies f r o m G F ( s ) . Our main

54

concern in the fractionally replicated designs is to maximize " d " so that

by running a factorial experiment of size s k - a, we bring out the wor-

thwhile information on all the k main effects and certain lower inte-

raction under the mild assumption that higher order interactions are

absent.

Definition 1.2.1: The partit ioning of a complete factorial experi-

ment s k is said to be of strength t if no main effect or t-factor or lower

order interaction is confounded.

Definition 1.2.2: A symmetrical factorial experiment s k is said to

be of the class (s k, s a) if it is part i t ioned in s a blocks, each of size

sn+l ,n+ l = k - d .

1.3. The Use of Finite Projective Geometry in the Construction of Arrays.

R. A. Fisher" [7,8] showed that the maximum number of factors

which can be accommodated in a symmetrical factorial experiment, in

which each factor is at s level, blocks are of size s" +1, without confou:z-

ding any main effect or two factor inr.eraction is given by the maximum number of distinct points in PG(n, s).

Bose [1] generalizing the Fisher's result proved the following

theorem.

Theorem 1.3: If mt(n + 1, s) denotes the maximum number of

factors, it is possible to accommodate is a symmetrical factorial

experiment in which each factor is at s level, each block is of size

s n+1, without confounding any degrees of freedom belonging to any

interaction involving t or lesser number of factors, then rnt(n + 1, s)

equals the maximum number of columns it is possible to take in an

(n + 1) - rowed matrix whose elements belong to GF(s), without

making any t columns linearly dependent .

In addition to the above result, we will employ the geometrical

method of construction developed by Bose and Bush [2]. The method

involves an investigation of k points in finite projective space PG(n, s) based on s distinct symbols o f GF(s), no t linearly dependent. Then

one multiplies the k X (n + 1) matrix so obtained, say C, by the matrix

55

Bn+ 1 consist ing o f all possible s n§ (n + 1) - tuples, incluiding the

(n + 1) - tuple consisting o f all zeros. The k X s "+1 p r o d u c t m a t r i x

will be an o r t h c , o n a l array o f s t rength t. The m a x i m u m n u m b e r o f

cons t ra in ts o f such a mat r ix clearly coincides wi th the m a x i m u m n u m -

ber o f po in ts which one can f ind in PG(n, s), no t l inearly d ep en d en t . 1.4 Confounding in Arrays of the type C X B

n + l ~

Consider a symmet r ica l fac tor ia l e x p e r i m e n t o f the class (2 k, 2 a )

o f 2 a blocks, each o f size 2 "§ where n + I = k - d. The (n + 1) in-

d e p e n d e n t t r ea tmen t comb ina t i ons o f (n + 1) factors , a,, a2 ,..., a n § ~

each at two levels, may be r ep re sen t ed in the fo rm of (n + l ) po in t s

El, E2 .... , En § ' where E i is a po in t in PG(n, 2) having ita coo rd ina t e

as un i ty and o the r coord ina tes are zero . The ma t r ix fo rmed by these

n + 1 l inearly i n d e p e n d e n t po in ts can be e x t e n d e d by in t roduc ing

k - (n + 1) fu r the r co lumns such that each addi t ional co lumn is a

c omb ina t i on o f (n + 1) e lements , each being a 0 or 1, so as to f o rm a

scheme o f n + 1 rows and k co lumns . Writing the fac tor n o t a t i o n ,

a,,+2, an.3 .... , a k above these k -- (n + 1) co lumns , the rows o f this

scheme give n + 1 i n d e p e n d e n t t r e a t m e n t combina t ions of k fac tors ,

each at two levels, and f rom these com b in a t i o n s a to ta l of 2 n*l _ 1

t r e a tme n t combina t ions o f k fac to r s can be ob ta ined exac t ly as descri-

bed earlier in the case o f (n + 1) factors . These 2 n*l _ 1 t r e a t m e n t

combina t ions along with a con t ro l t r e a t m e n t cons t i tu te the key-block

o f the design 2 k in blocks o f size 2 n § In what fol lows, we label the

first n + 1 independen t fac tors as Basic Factors while the remain ing

k - (n + 1) fac tors are designated as Supplementary Factors.

We shall now investigate the k - (n + 1) i n d ep en d en t in t e rac t ions

tha t are c o n f o u n d e d as a resul t o f ob ta in ing the key b lock in the

manne r descr ibed above. I f there is a un i ty at pth pos i t ion (p ~< n + 1)

in the c o l u m n below the f ac to r af (1 = n + 2,..., k), then an in t e rac t ion

conta in ing aj and all fac tors ap will be the i n d e p e n d e n t i n t e rac t ion

c o n f o u n d e d due to the i n t r o d u c t i o n o f the c o l u m n a.. This discussion 1

is due to Das [6] and is given here fo r pu rposes o f comple teness .

n F o r example , if n = 3, there are 4 i n d e p e n d e n t t r e a t m e n t com-

binat ions given below:

56

Basic Factors Supplementary Factor

al a2 a3 a4 aj 1 0 0 0 1

0 1 0 0 0

0 0 1 0 1

0 0 0 1 1

I f a new co l um n be low the f a c t o r a. wi th u n i t y in the p o s i t i o n s 1, 3 1

and 4 is i n t roduced , t h e n it fo l lows i m m e d i a t e l y tha t the i n d e p e n d e n t

in te rac t ion a 1 a3 a4 aj is c o n f o u n d e d .

When ] varies f r o m n + 2 to k, we get k - (n + 1) i n t e r a c t i o n s

( f rom the k - n - 1 c o l u m n s ) wh ich are c o n f o u n d e d in the k e y b ! o c k o f

size 2 ~ .1 involving k fac tors . The res t o f the 2 k - (~ +1 ) _ 1 i n t e r ac t i ons

c o n f o u n d e d are o b t a i n e d f r o m these k - n - 1 in t e rac t ions b y f o l l o -

wing the well k n o w n "Pr inc ip le o f Genera l i zed I n t e r a c t i o n " due to

Barnard. The fol lowing e x a m p l e due to Bose [1] gives the c o o r d i n a t e s

o f 8 po in t s in PG(3, 2), no th ree coll inear . These 8 po in t s are e x h i b i t e d

in the c o l u m n s of the fo l lowing m a t r i x : Basic Factors Supplementary Factors

al a2 a3 a4 as a6 a7 as f 0000 1 ] 1 0 0 1 0 1 1

0 1 0 1 1 0 1

0 0 1 1 1 1 0

A =

The t r e a t m e n t s are r ep re sen t ed by the finite po in t s o f PG(8,2) and the

design is o f the class t2 a , 2 4) ar ranged in 2 4 b locks each o f size 24 . The

in te rac t ions c o n f o u n d e d due to the i n t r o d u c t i o n o f the new fac to r s

as, a6, a7 and as are

a~a3a4as ; ala3a4a6; ala2a4aT ; ala2a3as.

The pr inc ip le of genera l ized i n t e r ac t i on yie lds 10 m o r e i n t e r a c t i o n s

be longing to f o u r - f a c t o r i n t e r a c t i o n whereas the r emain ing one be longs

to the e igh t - fac to r i n t e r ac t ion . This a ccoun t s fo r 15 degrees o f f r e d o m

c o n f o u n d e d .

57

In case it is desired to obtain a factorial design of the class (2 7, 2 3),

i. e., a 2 7 design in which a complete replication consists of 2 3 blocks,

each of 2 4 treatments, it is evident that seven degrees o f fredom are lost

in one replication. To obtain a design with seven factors, each at two

levels, in 2 3 blocks each of size 2 4 , in which no main effect or first or

second order interaction is confounded, we need to delete one column,

say the last, from the matrix A. Thus the columns of the matrix B

1 0 0 1 0 B = 0 1 0 1 1

0 0 1 1 1

give the coordinates of 7 points in PG(3,2), no three collinear. The

treatments in the design are now represented by the finite points of

PG(7,2). It is obvious that three degrees of freedom confounded belong

to the three interactions resulting from me introduction of the factors

as, a6 and a7 .while the Principle of Generalized Interaction will fix

the remaining four degrees of freedom that also belong to the four

factor interaction. All the seven degrees of freedom belonging to four

factor interaction are given below:

a 2 a 3 a a a 5 ; a ~ a 3 a 4 a 6 ; a a a 2 a 4 a T ; a ~ a ~ a s a 6 ; a ~ a a a s a T ; a 2 a 3 a 6 a 7 ;

a 4 a s a 6 a 7 .

1.5 Analysis of Orthogonal Arrays.

Consider a factorial experiment of the size 2 k. The treatment

combinations or assemblies will be written as a:fl ag2~3.., a~k or briefly

( x l , x : . . . . . x k ) where a i denotes the i ~ factor and x i gives the level of

i ~h factor. The various main effects and interactions are denoted by

(general mean), a i (main effect of i ~ factor), aia i or simply aq (inte-

raction between i n and jm factor). As in most practical situations, three

factor and higher order interactions are assumed negligible, we shall,

therefore, restrict ourselves to effects up to two factors and total nutn-

58

f / . ' ~ ber o f e f fec ts to be es t imated is thus v -- 1 + k +~.~J

Consider a matr ix A*(k , N) der ived by replacing - 1 for 0 in the

original array A = (N, k, 2, 5). Since A is an array o f s t rength five, so

is A * and in any 5-rowed submat r ix o f A*, a 5- tuple consist ing o f ele-

men t s 1 and - 1 appears X n u m b e r o f t imes. To es t imate v pa rame te r s ,

we cons t ruc t ano the r ma t r ix X(v, N) in which each row co r r e sponds

to the general mean, the nex t k rows co r re spond to the main e f fec t s

while the remaining{'~} rows tha t a c c o u n t for the in te rac t ions are ob- ~ , - - , J

ta ined as fol lows:

L e t a . be the row co r re spond ing to i th main e f fec t and a. be a n o t h e r t 1

row per ta ining to jth main e f fec t . T h e n the row . o r r e s p o n d i n g to the

in te rac t ion hi~ is ob ta ined by mul t ip ly ing the e lements in the row per-

taining to i th main e f fec t by the co r respond ing e lements in the row

relating t o jth main effect .

It may be easily verif ied tha t all the rows o f the m a t r i x X are

such tha t any two rows are mu tua l l y o r thogona l and each row excep

the first conta ins 16X l ' s and 16X - l ' s . The rows o f the ma t r ix X are,

there fore , the coeff ic ients o f a set o f mutua l ly o r t h o g o n a l cont ras ts .

Should it be evident that some in te rac t ion involving th ree or more

factors are not negligible and the i r con t r i b u t i o n in the e x p e r i m e n t is

o f some def in i te value, we may es t imate such in t e rac t ion by adding to

the rows o f matr ix X on ly those rows tha t d i rec t ly co r r e spond to the

fac tors involved.

The analysis o f the design m a y be carried ou t u n d e r the us,,al

a ssumpt ion o f the mode l i = k i = k

(1.9) Yi/...k = ~ + ~-a ai + ~ aia/ + eii...e i = 1 i = 1

where a i j . . . i c a s s u m e the value 0 or 1 and t~Ze j e r ror t e r m s eij...k are

mutua l ly i ndependen t and n o r m a l l y d i s t r ibu tea wi th m ean 0 and va-

r iance o 2 . The sum o f squares fo r main effects and in t e rac t ion m a y

be calculated by retaining on ly the assemblies p resen t in the array and

using p r o p e r divisors. In case o f Nas semb l i e s wi th k fac tors each at two

levels whe n array is o f s t rength grea ter than four , the analysis is as

below.

59

Analysis of Variance

Source

Main Effects

First Order First Order Interacions

Error

Degree of freedom

k

k ( k - 1)

2

obtained by subtraction

Total ....................... N - 1

2. Maximal ( k , t ) - sets.

Definition 2.1: A set of k points in PG(n , s), no t linearly depen-

dent, is called (k, t) - set.

Definition 2.2.: A (k, t) - set is said to be complete or maximal

if there exits no (k* , t) - set with k* > k. The number of points in

a maximal (k, t) - set denoted by m t ( n + 1, s).

Theorem 2.1 [4,9]: I f s ~< t, then mr( t , s) = t + 1.

Proof: Consider a set of t + 1 points in P G ( t - 1, s), no t depen-

dent. It is well known [5] that given a set of t + 1 points, no t of which

are linearly dependent , there exists a unique linear t ransformation which

take the set o f t + I points into a set of points Ei( i = 1 , 2 ..... t) , where

E i is a point having i-th coordinate as un i ty and E the unit point. This

provides a set of at least t + 1 points in P G ( t - 1, s) , no t dependent .

Hence, we have

(2.1.1.) k = m t ( t , s) >1 t + 1.

Suppose, if possible, that k > / t + 2 and then arrive at a contra-

diction. The coordinates o f t + 2 points may be exhibited in the co-

lumns of the following matrix:

60

(2 .1 .2)

m

1 0 0 0 .. 0 1 c11

0 1 0 0 .. 0 1 c21

0 0 1 0 .. 0 1 c31

0 0 0 1 .. 0 1 c41

~ . . . . . ~ . . . . . ~ 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 . . . . . ~ . . . . . . ~ 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6

0 0 0 0 . . 1 1 c S l

F r o m the nonvanish ing o f the d e t e r m i n a n t f o r m e d by any t - 1 co-

lumns f r o m the first t c o l u m n s and the ( t + 2) - th c o l u m n , it fo l lows

easily tha t c. 4: 0, i = 1, 2,.. . , s. Dividing out the last c o l u m n b y c1~, l 1

we can represen t the above m a t r i x in the fo l lowing f o r m : n

1 0 0 0 . . 0 1 1

0 1 0 0 .. 0 1 d21

0 0 1 0 .. 0 1 d31

0 0 0 1 .. 0 1 d41 (2 .1 .3)

~ ~ 1 7 6 1 7 6 1 7 6 1 7 6 . . . . . . . ~ . . . . . . . . �9 . . . . . . . . . . . . . . . . . . . . . . . ~

0 0 0 0 . . 1 1 d S l

where d ~ 0, i = 2, 3,... , s. The e l e m e n t s d. c a n n o t be u n i t y since, I I / 1

a p p r o p r i a t e select ion o f t -- 2 c o l u m n s f r o m the first t c o l u m n s and

the last two co lumns p rov ides a vanish ing d e t e r m i n a n t . I t fo l lows by

similar reasoning that since the d e t e r m i n a n t f o r m e d b y the f irs t , any

set o f t - 3 co lumns f r o m the 2nd , 3rd ..... t - th co lumns , and the

last two co l um ns does no t vanish , the e l emen t s d=l, dal ..... dsl are

all d i f f e ren t and m a y , t he r e fo re , be ident i f ied wi th the r ema in ing s - 2

nonze ro e l e m en t s o f GF(s ) . Since the re are s - 1 places to fill and

s - 2 < t - 1, this is imposs ib le . Th is c o n t r a d i c t i o n p roves the asser-

t ion tha t a ma t r ix wi th t + 1 c o l u m n s canno t be c o m p l e t e d to t + 2

co lumns .

T h e o r e m 2 .2 . : mt(t + 1, 2) = t + 2 for t > 3.

Befo re prov ing this t h e o r e m we wish to r e m a r k tha t the ex i s t ence

o f 7 p o i n t s for t = 2 and tha t o f 8 nonco l l i nea r po in t s fo r t = 3 are

consequences o f the resul ts es tab l i shed earl ier b y F isher [7] and Bose

61

[ 1 ]. The assumpt ion t > 3 is t he re fo re essential.

Proof: Not ice that any max ima l (k, t ) - set m u s th av e t + 1 l inear ly

i n d e p e n d e n t vectors . Choose , i f possible, a maximal l inearly indepen-

den t subset o f the maximal (k, t ) - s e t w h i c h m a y b e d e n o t e d by the set

o f vectors Xl , x_2, .... _x , where u < t + 1. Clearly, there exists a vec to r

_y which c a nno t be wr i t t en as a l inear c o m b i n a t i o n o f the u vectors cho-

sen above. This gives us a set o f k + 1 vec tors con t rad ic t ing the assump-

t ion that we had a maximal set to s tar t with.

Cons ider a set o f t + 1 po in t s spanning a t -d imensional space.

There exists a un ique linear t r an s fo rma t io n which would conver t these

poin ts to El , E2 , . . . , Et+ 1 w h e r e E i ( i = 1, 2,.. . , t + 1) is a poin t having

i=th coord ina t e as un i ty while o t h e r coord ina tes are zero. A n y p o in t

added to this set o f t + 1 poin ts mus t have at least t ones if no t po in t s

in the augmen ted set are to be l inearly dependen t . I f the new p o in t

has all t + 1 ones, then no fu r the r po in t can be added. We have t h e n

a set o f t + 1 points , no t + 1 o f which are l inearly d ep en d en t , and

this in turn implies no t are also l inearly dependen t . Cons ider n ex t the

case where the new po in t has exac t ly t coord ina tes as un i ty . Wi thou t

any loss o f general i ty , we m a y assume tha t

(2 .2 .1)

It fo l lows tha t

(2 .2 .2) ai, t§ 3

ai, t . 2 = 0, a. , t .2 = 1 f o r j 4=/.

= 1 , a , t + 3 = 0 f o r j = ~ i, ar , t . 3 = 1 f o r r = / : i , j .

But there two new po in t s fo rm a l inearly d e p e n d e n t set with t - 2

po in t s appropr ia t e ly chosen f rom E'is. QED

= I t + 4 for t = 4 and t = 5

/

Theorem 2.3: mt(t + 2, 2) + 3 f o r t ~ > 6.

,

Proof: The fact tha t a ( t - 1) - d imens iona l pro jec t ive space based

on two e lements 0 and 1 conta ins exac t ly t + 1 points , no t d e p e n d e n t ,

is a consequence o f ou r previous ly establ ished resul t in T h e o r e m 2.1.

Exac t ly 3 t-flats pass t h rough this ( t - 1) - flat. T h e o r e m 2.2 assures

62

that no t-space can contain more than t + 2 points. Hence, each of

the 3 t-flats can contain at most one point that does not lie in

(t - 1) - space. Thus

mt(t + 2,2)<~(t + 1 ) + 3 = t + 4 .

Rao [ 13] showed the existence o f 8 points in PG(5, 2), no four co-pla-

nar. To show that a bound can be attained for t = 5, we give below an

example of 9 points in the columns of the following matrix, no five

dependent.

1 1 0 0 0 0 0 0 0 1

0 1 0 0 0 0 0 0

0 0 1 0 0 0 0 1

0 0 0 1 0 0 0 1 i l 0 0 0 0 1 0 0 1

0 0 0 0 0 1 0 1

0 0 0 0 0 0 1 1

We have remarked earlier that new points added to E; (i = 1,2,..., t + 2)

must have at least t coordinates equal to unity if any set of t points

is to be linearly independent. Without any loss of generality, we may

represent additional two points as follows:

u' = ( 0 , 0 , 1, 1,..., 1, 1, 1)

v ' = ( 1 , 1, 1, 1,..., 1 , 0 , 0 ) so that

u ' + v ' = ( 1 , 1, 0, 0,..., 0, 1, 1). m

!

Now it follows easily that El, E~., Et+l, Et, 2 along with the vec to r su

s (t + 2, 2) and v form a dependent set. We, therefore, conclude that m t = t + 3 for t l > 6 . F o r t = 2, r n = ( 4 , 2 ) = 15 and f o r t = 3 , rn3 (5, 2)

= 16 and as such the assumption t > /4 is necessary.

Theorem 2.4: ms (8, 2) = 12

Proof: Lek k = mt(n + 1, s) have the usual meaning. Rao [13]

has shown that, rn4 (7, 2) = 1 1. Later, Seiden [ 14] has established the

63

same result. Since an increase in the value of t and n by one can result in

the corresponding increase in k by at most one, it follows that ms (8,2)

cannot exceed 12. We now demostrate the existence of 12 points in

PG(7, 2) in the columns of the following matrix satisfying the cond i -

tion that no five of these are linearly dependent.

1 0 0 0 0 0 0 0 t 0 0 1

0 1 0 0 0 0 0 0 1 O 1 1

0 0 1 0 0 0 0 0 1 0 1 0

0 0 0 1 0 0 0 0 1 1 O 1

0 0 0 0 1 0 0 0 1 1 O 0

0 0 0 0 0 1 0 0 1 1 1 0

0 0 0 0 0 0 1 O 1 1 1 1

0 0 0 0 0 0 0 1 0 1 1 1

3. Const ruc t ion of Arrays, t = 5, s = 2.

Theorem 3.1: For an orthogonal array (32, k, 2, 5), the maxi-

mum value of k is six.

For arrays of unit weight and t ~> 2, the existence of an array

(2 t, t + 1, 2, t)was established by Bush [4]. Consider any t + 1 tuple

and adjoin it to all the 2 t - 1 (t + 1) - tuple that differ from it by

an even number of elements. I t is readily seen that for t odd, the

columns of the array consist of either an even or an odd number of

both zeros and ones. The array with the first column consisting o f all

zeros will be denoted b y D whereas D* will represent an array with an

odd number of both zeros and ones. For t = 5, the two fo,-ms of the

array (32, 6, 2, 5) are given below:

64

D:

0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1

0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 1

0 1 0 1 1 1 0 1 1 1 0 0 0 1 1 1 0 1 0 0 0 1 0 0 0 1 1 1 0 0 0 1

0 1 1 0 1 1 1 0 1 1 0 1 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 1 0 1

0 1 1 1 0 1 1 1 0 1 1 0 1 0 1 0 0 0 0 1 0 0 0 1 0 0 1 0 1 0 1 1

0 1 1 1 1 0 1 1 1 0 1 1 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 1 1

D*:

O l l l l l O 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0

I 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 0

1 1 0 1 1 1 0 1 1 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 0

1 1 1 0 1 1 1 0 1 1 O 1 1 0 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 1 O0

1 1 1 1 0 1 1 1 0 1 1 0 1 0 1 0 1 0 1 0 1 0 0 1 0 0 0 0 0 0 1 0

i 1 1 1 1 0 1 1 1 0 1 1 0 1 0 0 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 1

It is readily seen that there is

linearly dependent, that can be

essentiallyone set o f six points, no five

exhibited as follows:

1 0 0 0 0 1

0 1 0 0 0 1

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

1

1

1

- (5 , 6 ) .

Multiplying the transposed 6 X 5 matrix by tile 5 X 2 s matrix consis-

ting of all possible 5-tuples of two elements 0 and 1, one obtains the

required array /9. The fact that six is the maximum number of cons-

traints follows from theorem 1.2.

Theorem 3.2: For an orthogonal array (64, k, 2, 5), maximur,

value of k is seven.

Necessary conditions (1.7) and (1.8) reduce to the following set of six linear equations:

65

rzi0 = 1 + r/i6 /7il = 6 - 6rti6 rti2 = 15 + 15rli6

/7i3 = 20 -- 20rti6 rti4 = 15 + 15ni6 his = 6 -- 6 n i 6 .

Since there is no negative solut ion for tTijts, it fol lows that ni6 = 0 or 1.

This gives the fol lowing two solut ions.

T a b l e I

/ ' / i0 /2i l / '/i2 / ' / i3 / ' / i4 ] '/is n i 6

6.1 1 6 15 20 15 6 0

6.2 2 0 30 0 30 0 1

The solut ion 6.1 reflects a mat r ix o f the size 6 X 2 6 o f all possible

6-tuples o f two e lements 0 and is o f the nature

D D *

while solut ion 6.2 may be represented in the fol lowing fo rm

D D,

For k = 7, we have the fol lowing set o f equat ions .

T/iO ~--- Y/i6 "~- Y/J7

n i l = 7 -- 6ni6 -- 35n i7

/ ' li2 = 1 5 n i 6 "1- 8 4 n i 7

ni3 =-- 35 -- 20ni6 -- 105ni7

l"li4 = l 5 n i 6 '~ 8 4 / ' / i 7

his = 21 -- 6ni6 - 21ni7

if n . . ' s are to assume non-negat ive values, it fol lows easily that ni7 = 0. U

The above equa t ions yield the fo l lowing two solut ion with seven c o n s -

traints:

66

Table II

Ylio l'li 1 hi2 h i3 t7i4 h i s fli6 l'li7

7.1 1 1 15 15 15 15 1 0

7.2 0 7 0 35 0 21 0 0

The solutions 7.1 and 7.2, which are extensions of 6.2 and 6.1

respectively, may conveniently by represented as follows:

7.1 7.2

D D D D *

0 1 0 1

where 0 or 1 below the array D or D * means that to each of the

columns of the array the same element, either 0 or 1, is added.

That there are seven points in PG(5 , 2) satisfying the basic con-

dition that no five of these are linearly dependent follows from Theo-

rem 2.2. These points may include either (I , 1, 1, 1, 1, 0)' or (1, 1, 1,

1, 1, 1 ,)'. These two sets of seven points are exhibited below:

cI=

1 0 0 0 0 0 1

0 1 0 0 0 0 1

0 0 1 0 0 0 1

0 0 0 1 0 0 1

0 0 0 0 1 0 1 0 0 0 0 0 1 0 (6, 7)

c;=

I 0 0 0 0 0 1

0 1 0 0 0 0 1

0 0 1 0 0 0 1

0 0 0 1 0 0 1

0 0 0 0 1 0 1 0 0 0 0 0 1 1 (6, 7)

Multiplying each of the transposed 7 X 6 matrix C, and (72 by the

6 X 26 matrix of all possible 6-tuples, we obtain the desired arrays

7.1 and 7.2, respectively. It remains to be shown that none of these arrays can be extended

further. The necessary equations (1.7) and (1.8) may be simplified to

yield the following equations with nine variables.

67

n io -- 42 +

n i l = 27 --

ni2 -- 56 +

n i3 = 1 1 3 -

n i4 - 70 +

ni5 = 56 -

ni6 -t- 6 n i 7 + 2 1 n i a

6n i6 -- 3 5 n i 7 - - 120nia

1 5 n i 6 q- 84ni7 + 280his

20ni6 --105/7i7 - - 3 3 6 n i 8

15ni6 + 7 0 n i 7 + 210nis

6n i6 - - 2 l n i 7 - - 5 6 n i 8

(i)

(ii)

(iii)

(iv)

(v)

(Vi)o

In view o f the fact tha t n. . 's can assume on ly non-negat ive values, equa- U

t ion (ii) implies tha t n i t -=- hi8 = 0 and ni6 ~< 4, while equa t ion (v) re-

quires tha t h i 6 ~ 5. This con t r ad i c t i on concludes the p r o o f tha t maxi-

m u m n u m b e r o f cons t ra in ts is seven.

T h e o r e m 3.3: F o r an o r t hogona l array (96, k , 2, 5), the m a x i m u m

value o f k is six.

In this case X = 3 and hence odd. F u r t h e r m o r e , X ~< t - 1 = 4,

and tha t the m a x i m u m n u m b e r o f cons t ra in ts is t + 1 = 6 is a conse-

quence o f ou r previously stated T h e o r e m 1.2.

This array can be expressed as a jus tapos i t ion o f three arrays

(32, 6, 2, 5), each o f un i t weight.

T h e o r e m 3.4: F o r an o r t h o g o n a l ar ray (128, k, 2, 5), the maxi-

m u m value o f k is nine.

F o r k = 6, 7, 8 and 9, we have the fo l lowing set o f l inear equa-

tions:

k = 6 : nio = 1 + n i6

H i l ~ l 'li 6

hi2 = 15 + 1 5 h i 6

n i3 = 6 0 - - 2 0 n i 6

Hi4 = 15 + 15ni6

h i s = 18 - - 6 n i 6

68

/<- = 7 " nio = -- 6 + hi6 + ni 7

n i l = 4 9 - 6ni6 - 3 5 n i 7

r7i2 = - - 8 4 + 15ni6 + 8 4 n i 7

F/i3 = 1 7 5 - - 2 0 n i 6 -- 105F/ i7

h i4 = -- 7 0 + 1 5 n i 6 + 7 0 h i 7

fli5 = 63 - 6 n i ~ - 2 1 r

k = 8 nio = -- 2 7 + ni6 + 6ni7 + 2 1 h i s

nia = 1 6 8 -- 6ni6 -- 3 5 n i 7 -- 1 2 0 h i s

hi2 - 3 9 2 - F 1 5 n i 6 -F 8 4 n i 7 + 2 8 0 n i s

ni3 = 5 6 0 - - 2 0 n i 6 - 1 0 5 n i v - 3 3 6 h i s

Hi4 -- 3 5 0 - - ~ 1 5 n i 6 + 7 0 n i 7 + 2 1 0 h i s

fli5 = 1 6 8 -- 6ni6 -- 2 1 n i 7 -- 5 6 t l i a

k = 9 : nio -- 8 0 + ni6 + 6ni7 + 2 1 h i s + 5 6 n i 9

;lil = 4 7 7 - 6ni6 - 3 5 n i 7 - 1 2 0 h i s - 3 1 5 n i 9

hi2 = - 1 1 5 2 + 1 5 n i 6 + 8 4 n i 7 + 2 8 0 n i s + 7 2 0 n i 9

hi3 = 1 5 1 2 - 2 0 p l i 6 - 1 0 5 n i , 7 - 3 3 6 h i s - 8 4 0 n i 9

r/i4 = - - 1 0 0 8 + 1 5 r / i 6 7 0 n i 7 -- 2 1 0 n i s -- 5 0 n i 9

rzis - 3 7 8 - 6ni6 -- 2 t n i 7 - 5 6 n i s - 1 2 6 n i 9

Th:: i 'oliowin~ tab l : elves all so lut ions for k = 6, 7, 3 and 9.

69

nio nil

T a b l e I I I

ni2 ni3 hi4 his ni6 ni t

6.1 1 18 15 60 15 }8 0

6.2 2 12 30 40 30 12 1

6.3 3 6 45 20 45 6 2

6.4 4 0 60 0 60 0 3 0

7.1 0 13 6 55 20 27 6 0

7.2 1 7 21 35 35 21 7 0

7.3 2 1 36 15 50 15 88 0

7.4 0 14 0 70 0 42 0 1

7.5 1 8 15 50 15 36 1 1

7.6 2 2 30 30 30 30 2 1

8.1 0 6 13 20 55 6 27 0

8.2 1 0 28 0 70 0 28 0

8.3 0 7 7 35 35 21 21 1

8.4 1 1 22 15 50 15 22 1

8.5 0 8 1 50 15 36 15 2

8.6 1 2 16 30 30 30 16 2

9.1 0 4 6 17 52 3 38 7

9.2 0 5 0 32 32 18 32 8

9.3 1 0 9~ 27 27 27 27 9

9.4 1 1 3 42 7 42 21 i0

9.5 1 0 10 21 42 7 42 3

9.6 1 1 4 36 22 22 36 4

/'/ib /gi9

0

0

0

0

0

0

0 0

0 0

0 0

0 0

1 0

1 0

Arrays admitting each of the solution for k = 6 and k = 7 can be

effectively constructed. These will not be exhibited in detail as these

constitute subarrays of some larger arrays to be constructed later. Ho-

wever, the following table provides information in respect of subarrays

of the arrays with seven constraints.

70

Table IV

Seven-rowed arrays

7.1

7.2

7.3

7.4

7.5

7.6

Six-rowed subarrays

one subarrays of the form 6.1

l six of the form subarrays 6.2

seven subarrays of type 6.2

l six subarrays of the type 6.2

one subarrays of the type 6.3

seven subarrays of the type 6.2

l six subarrays of the type 6.2

one subarrays of the type 6.3

l one subarravs of the type 6.4

six subarrays of the type o.2

On can, in fact, eight constraints. The

7.1 and can be represented as follows:

D D*

0 0

0 1

The first two rows constitute the

rowed arrays along with their source

construct arrays satisfying all solutions with solution 8.1 is an obvious extension of arrays

D* D*

1 1

0 1

solution 7.1. The remaining eight-

of origin are given below:

D D D* D*

8.2: 1 0 1 0

1 0 0 1

D D D* D*

8.3: 0 0 1 1

0 1 0 1

D D D D* 8.4: 1 1 0 0

0 1 0 1

71

D D D D* 8.5: 0 0 1 1

0 1 0 1

D D D D

8.6: 0 0 1 1 0 1 0 1

It may be observed that first two rows of the above eight-rowed arrays

const i tute seven-rowed or thogonal arrays 7.2, 7.4, 7.3, 7.5 and 7.6,

respectively. We, however, admit that the above enumerat ion is not

exhaustive since array 7.2 can also be extended to satisfy the solut ion

8.1. Fur ther investigation reveals that array 7.2 can in fact be ex tended

to all the eight-rowed arrays. Again, the array 7.5 may be ex tended

to the solution 8.4. The complete enumera t ion does not seem to be

worthwhile because of the unique structure of (128, 9, 2, 5) with nine constrainst.

One may also exhibit the set of 8 points in PG(6 2), no five de-

pendent , in the following three matrices.

1 0 0 0 0 0 0 1 I 0 1 0 0 0 0 0 1

O 0 1 0 0 0 0 1 , /0 0 0 0 1 0 0 0 1 C 2 = 0

0 0 0 0 1 0 0 1 l i 0 0 0 0 0 1 0 0

0 0 0 0 0 0 1 0

G =

0 0 0 0 0 0 1

1 0 0 0 0 0 1

0 1 0 0 0 0 1 0 0 1 0 0 0 1

0 0 0 1 0 0 1

0 0 0 0 1 0 1

O 0 O O 0 1 0

1 0

0 1 O 0

G = O 0 O 0

O 0

O 0

The premult ipl icat ion

0 0 0 0 0 1 0 0 0 0 0 1

1 0 0 0 0 1

0 1 0 0 0 1 0 0 1 0 0 1

0 0 0 1 0 1 0 0 0 0 1 1

o f t h e m a t r i x 7 X 2 7 of all possible seven-

72

tuples by the. transpose o f each o f the above matr ices yields so lu t ions

8.6, 8.3 and 8.2, respect ively.

I_emma 3.4.1: The solut ions 9.1, 9.2, 9.4, 9.5 and 9.6 do n o t re-

present o r thogona l arrays.

Proof: It need be observed tha t arrays 9.4 and 9.5 are o b t a i n e d

f rom each o the r by an in terchange o f two e lements 0 and 1. It, there-

fore , suffices to show that the so lu t ion 9.4 is no t o r thogona l .

An array of type 9.4 has one co lu m n of all l's and one c o l u m n

having one coincidence with the co lum n of all o 's. Delet ing the row

in which this coincidence occurs , we get an e ight- rowed suba-

tray with nio = 2, which is impossible. Tile fact that 9.6 is no t o r tho -

gonal fol lows by tile same a rgument .

To shows that no array exists sat isfying solut ion 9.1, observe tha t

four co lumns having one co inc idence must have this co inc idence in

d i f fe ren t rows in order to e l iminate eight or more co inc idences amongs t

themselves. Deleting one of these rows in which the co inc idences occurs ,

we obta in an eight-rowed subarray with ni~ = 1 and nio ~> 3, which is

impossible.

The solut ion 9.2 is not an ex tens ion o f arrays 8.2. 8.4 or 8.6,

since any extens ion of these arrays must have hi0 + ni~ ~< I, ,~'~ and 3,

respect ively , but nio + t~i~ = 5. Arrays 8. I, 8.3 and 8.5 have rtil equa l to

6, 7 and 8, respect ively, while nil -- 5 in the so lu t ion 9.2. If any o f the

arrays 8.1, 8.3 or 8.5 could be e x t e n d e d to 9.2, then five o f the c o -

lumns having one coincidence in the e igh t rowed arrays have to have 1

added in the ex tended row while the remain ing co lumns would have

0 added in the ex tended row. Thi, s gives ni2 > O, b u t hi2 = 0 in the so- lu t ion 9.2. QED

Lemma 3.4.2: Arrays 8 .1 ,8 .2 , 8.3, 8.4 and 8.5 cannot be e x t e n d e d .

Proof: The fact tha t 8.1, 8.3 and 8.5 can n o t be ex t en d ed to the

un ique array 9.3 fol lows readi ly since in all the above e ight - rowed

arrays n. o = 0, while 9.3 has nio = 1. In case array 8.2 could be ex ten-

ded, t hen its ex tens ion would have to have nio + ni~ + ni2 + nia <~

73

29, but these coincidences add to 37 in the solut ion 9.3. To show

that 8.4 is no t a subarray o f the un ique array 9.3, it suffices to observe

that tl-.,e delet ion of any row of 9.3 yields an eight-rowed subarray wi th

nio = 1 and nil = 2 while 8.4 has ni, = 1. QED

If may be no ted that there are nine points in PG(6,2), no five

linearly dependent . It can be easily shown that if one includes 2 more

points in the set o f seven points forming the ident i ty matr ix , then there

is jus t one way of complet ing this set to nine points up to inter-

changing the role o f the coordinates . A set o f nine points is exhibi ted

in the theorem 2.3. The premul t ip l ica t ion o f the mat r ix 7 • 2 7 of all

possibla seven-tuples by the transpose o f tha t mat r ix yields the solu-

t ion 9.3.

It remains to be shown that the un ique array 9.3 cannot be exten-

ded further. I f this array could be ex tended , then any extension o f

9.3 mus t have ni9 = ni~ 0 = 0. There are only two solut ions provided

by the equat ions (1.7) to (1.8) which are enumera ted below:

nio nil rli2 1"1i3 17i4 gli5 ni6 F/i7 /7i8 /7i9 /7il 0

10.1: 1 0 0 30 15 36 15 30 0 0 0

10.2: 1 0 1 24 30 16 30 24 1 0 0

An array of type 9.3 has the fol lowing s t ructure ; it has one c o l u m n o f

all l ' s and nine columns having two coincidences shown below:

1 0 1 1 I 0 1 1 1 1

1 1 0 1 1 0 1 1 1 1

1 1 1 1 1 1 0 0 1 1

1 1 1 1 1 1 0 1 0 1

1 1 1 0 1 1 1 0 1 0

1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 0

1 1 1 0 0 1 1 1 1 1

1 0 0 1 1 1 1 1 1 1

If the array 9.3 could be ex tended to s a t i s ~ solut ion 10.1, then all

74

the nine columns of ni2 have to have 0 added in the extended row. But

this provides second and third columns having eight coincidences

amongst themselves. This is impossible since n;8 = 0 in the solution

10.1.

An array with solution 10.2 is not an orthogonal array. This fo-

llows from the fac. solution 10.2 has one column of all l's and one

columns having two coincidences with the i th column assumed to

consist of all 0's. If we delete one of the rows in which a zero appears,

= 1 and n. = I But 9.3 has we have a nine-rowed subarry with n i o z l �9

nil = 0. A contradiction.

This concludes the proof of the theorem.

4. C o n s t r u c t i o n o f A r r a y s , s = 3, t = 5.

To construct an orthogonal array (243, 6, 3, 5) of unit weight,

we note that there is essentially one set of six points inPG(4, 3), no

five linearly dependent, which may be exhibited as follows:

/lS 1] 1 1 1

A pemultiplication of this matrix by the 5 X 3 s matrix consisting of

all possible five-tuples of three elements 0, 1 and 2, yields the

desired array. That "qx is the maximum number of constraints is a

consequence of Theorem 2.1.

To construct an array (729, 12, 3, 5) of weight three, we may

~,ecall Rao's Theorem (1.1) which assures us that the maximum value

of k is given by

The equality holds in the above relation for k = 12. An example of

12 points in PG(5, 3), no five linearly dependent, is given below.

75

i i 0 0 0 0 0 0 1 1 11 0 1 0 0 0 0 1 0 1 2 1 0 0 1 0 0 0 1 1 0 1 2 ~ 0 0 0 1 0 0 1 2 1 0 2 0 0 0 0 1 0 1 1 2 2 0 0 0 0 0 0 1 1 2 2 1 1

Multiplying the transpose of this matrix (12 • 6) by the 6 • 36 matrix of all possible six-tuples consisting of three elements O, 1 and, 2, we get the required array

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[10] GULATI, B.R. (1971) On maximal (k,t) - sets. Ann. Statist. Math., 23(2), 279-292. [11 ] HOTELLING, H. (1942). Some improvements in weinghtingand

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77