Vol.ll No.3 ACTA MATHEMATICAE APPLICATAE SINICA July, 1995

A SURVEY OF ORTHOGONAL ARRAYS

OF STRENGTH TWO

LIU ZHANGWEN (~] .~ ~ ) *

(Institute of Applied Mathematics, the Chinese Academy of Sciences, Belting 100080, China)

FuJII YOStilo ( ~ ~ ~ 5~)

( Okayama University # Science, 700, Japan) and

(International Institute for Natural Science.s, Kurashiki, 710, Japan)

A b s t r a c t

The purpose of this paper is to survey the construction of orthogonal arrays of strength two by using difference sets. Some methods for constructing difference set D(2p,2p,p,2), where p is a prime or a prime power, axe given. It is shown that the Kronecker sum of a differ- ence set D(Alp,kl,p,2) and an orthogonal array (A~p2,k2,p,2) leads to another orthogona] ar- ray (AIA2ps,k~k2+l,p,2). This enables us to construct orthogonal arrays [2p ~+~, l+2(p+p2+.-.+

where p is a prime or a prime power.

Key words. Orthogonal array, difference set, kronecker sum, aAqne resolvable BIB design.

1. Introduct ion and Summary

An N × k mat r ix A with p elements is called an orthogonal array of s t rength t, size N, k constraints and p levels, denoted by (N, k,p, t), if each N × t submatr ix of A contains all possible 1 × t row vectors with the same frequency A. The number A is called the index of the array. Clearly N = Apt.

The concept of orthogonal arrays was first introduced by Rao! 14]. He discussed the use of these arrays of s t rength t as fractionally replicated plans for symmetr ical factorial experiments which permit the est imation of the main effects and interactions up to an order [~ - 1] when higher order interactions are negligible, where [x] is the largest possible integer not exceeding x. In this paper we consider the case t = 2. So the array may be denoted by (Ap 2, k,p, 2). The op t imum multifactorial designs constructed by Plackett and Burman [13] are essentially orthogonal arrays of s t rength two.

Received June 13, 1992.

*This work was carried out while the author was a visiting professor at the International Institute for Natural

Sciences, Kurashlld 710 Japan, during Oct. 1990-July 1991.

No.3 A SURVEY OF ORTHOGONAL ARRAYS OF STRENGTH TWO 309

It is known that orthogonal arrays of strength two can be regarded as natural gener- alizations of orthogonMM Latin squares. Suppose there is a set of p - 1 mutually orthogonal p × p Latin squares. Then an orthogonal array

(p2,1 + p , p, 2), (1.1)

whose k = 1 + p constraints represented by the row, column, and p - 1 squares attaining the known upper bound for the maximum possible number (see [3]), can be constructed.

It is also known that the method of differences which has been used for constructing orthogonal arrays of strength two is due to Bose and Bush[ 3] . They constructed a difference set D(2 • 3, 2 • 3, 3, 2) by trial and error methods, which generated an orthogonal array (2 • 32, 2 .3 + 1, 3, 2) of strength two. The orthogonal array (18, 7, 3, 2) has been first constructed by Su rman (see "Added in Proof" of [13]) by trial and error.

Masuyama [1°'11], Shrikhande [17], Liu [9], Jiang Is] , Xu [191 and Xiang [is] have shown that difference set D(2p, 2p, p, 2) exists where p is a prime or a prime power.

In [4-7, 12, 20], the existence, construction and classification of orthogonal arrays with parameters (A2 t, t+3, 2, t) of A < 2 t have been investigated completely. Recently, Yamamoto, Fuji and Mitsuoka [21] are investigating the existence, construction and classification of 3 level orthogonal arrays with parameters (2.32, m, 3, 2) of 2 < m g 7.

In this paper the methods for constructing D(2p, 2p, p, 2), where p is a prime or a prime power, by Liu [91 , Jiang Is], Xu [19] and Xiang [lsl will be cited, because they have been published in Chinese and are not so well-known abroad, the methods for constructing orthogonal arrays of strength two are also given.

2 . C o n s t r u c t i o n o f D i f f e r e n c e S e t s

Let M denote a module of order p and let D be a 2p x 2p matrix with elements from M. Then D is called a difference set D(2p, 2p, p, 2) if the ordered differences arising form any two columns of D contain all the elements of M exactly two times.

T h e o r e m 1 [9]. Let p = 6n - 1 be a prime. Then the 2p x 2p matrix

D6n-1 [(alj) (bij) ] = (c,j) (d,j)] is a difference set D(2p, 2p, p, 2) in the residues (rood

a i j = i j , b i j = i ( i + j ) , c i j = ( i + j ) j ,

and i , j = 0,1, . . . , p - 1= 6 n - 2. E x a m p l e 1. Let n = 1. Then a D(10, 10, 5, 2)

D 5 =

p) where

d~j = - ( i 2 + ij + j2) /3 ,

is as follows:

0 1 4 4 1 0 3 2 2 3 0 2 1 2 0 3 4 1 4 3 0 3 3 0 4 2 1 1 2 4 0 4 0 3 3 2 4 2 1 1 0 0 2 1 2 3 3 4 1 4.

"0 0 0 0 0 0 0 0 0 0" 0 1 2 3 4 1 2 3 4 0 0 2 4 1 3 4 1 3 0 2 0 3 1 4 2 4 2 0 3 1 0 4 3 2 1 1 0 4 3 .2

310 ACTA MATHEMATICAE A P P L I C A T A E SINICA Vol . l l

T h e o r e m 2 [9] • Let p = 6n ÷ 1 be a prime. (i) For p = 6n + 1 ~ 1, 4 (rood 5), the 2p x 2p mat r ix

D6,~+I--- L(ci~) (dij)J

is a D(2p, 2p, p, 2) where

~ j = i j , b~j = - i ( i - j ) ,

and i , j = 0 ,1 , . . - , p - 1 = 6n.

cij = (i ÷ j ) j , dis = (_i2 ÷ i j + j 2 ) / 5 ,

(ii) For p = 6n ÷ 1 ~ 1, 2, 4 (rood 7), the 2p x 2p mat r ix

6~+x= k(c,A (d,AJ

is a D(2p, 2p, p, 2) where

aij = i j , bij = - i ( i - j ) ,

and i, j = 0 ,1 , - - - ,p - 1 = 6n.

cij - . ( i - 2 j ) j , di j = (i2 _ i j + 2j2)/7,

I R e m a r k . Let p = 2m + 1 instead of p ---- 6n -b 1. I t holds for bo th D6,,+I and D6,,+ I. E x a m p l e 2. Let m = 1. Then a D(6, 6, 3, 2) is as fol]ows:

T h e o r e m 3 Is]. Pu t

Then the 2p x 2p matr ix

"0 0 0 0 1 2 0 2 1

D3 = 0 1 1 0 2 0

. 0 0 2

Let p be an odd prime

~ - 1 4~ (rood p),

0 0 0 2 0 1 2 1 0

0 2 2 1 2 1 1 1 2

and let 6 be a quadratic non-residue (mod p).

- a6 ((mod p).

[(a,j) (b,j)] Dj - - L(c,j) (dij)J

is a D ( 2 p , 2p, p, 2) where

a~j = i j , b~j = i ( i + j ) ,

and i, j = 0 , 1 , . . . , p - 1.

cij --= (i + a j ) j , di j = 5(i 2 + i j ) ÷ ~j2,

E x a m p l e 3. Let p = 6n .- 1. As shown in [9], - 3 is quadratic non-residue (mod p), therefore - 1 / 3 is also a quadratic non-residue. Put t ing 5 -- - 1 / 3 , we have

~5 = ;3 = - 1 / 3 (mod p), a = 1 (mod, p).

Clearly, entries of D j for p = 6n - 1 are same as those of D 6 . - 1 of T h ~ r e m 1.

No.3 A SURVEY OF ORTHOGONAL ARRAYS OF S T R E N G T H T W O 311

T h e o r e m 4 [19l. the 2p x 2p mat r ix

Let p be an odd prime and a a quadrat ic non-residue (mod p). Then

[ ( a ~ j ) ( b l j ) ] D x u = (cij) (dij)

is a D(2p, 2p,p, 2) where

aij = 2ij , bij = i(i + 2j),

and i, j = 0 , 1 , . . . , p - 1.

cij = (2 i + v~- l J ) dij = a(i2 + 2ij) + (c~ - 1)j 2,

E x a m p l e 4. Let p = 4n - 1. As is well-known, - 1 is a quadratic non-residue (mod p). Pu t t ing n = 2 we have the following

D x , ( 1 4 , 1 4 , 7 , 2 ) =

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 4 6 1 3 5 1 3 5 0 2 4 6 0 4 1 5 2 6 3 4 1 5 2 6 3 0 0 6 5 4 3 2 1 2 1 0 6 5 4 3 0 1 2 3 4 5 6 2 3 4 5 6 0 1 0 3 6 2 5 1 4 4 0 3 6 2 5 1 0 5 3 1 6 4 2 1 6 4 2 0 5 3

0 2 1 4 4 1 2 0 5 6 3 3 6 5 0 4 5 3 5 4 0 6 2 1 3 1 2 6 0 6 2 2 6 0 5 3 4 1 1 4 3 5 0 1 6 1 0 3 3 5 4 6 4 5 2 2 0 3 3 0 1 6 1 5 2 2 5 4 6 4 0 5 0 6 2 2 6 3 5 3 4 1 1 4

0 0 4 5 3 5 4 6 6 2 1 3 1 2

T h e o r e m 5 [17'1s1. I f Di = D(A~p,k~,p, 2), i = 1,2 are two difference sets with elements from M, then D = D1 @ D2 is also a difference set D()h)~2p 2, klk2 ,p , 2) in the residues (rood p) where @ denotes the Kronecker sum.

I t is known from [9] that the multiplication table of M is a difference set D(p ,p ,p , 2). Hence for p = 2 we have the following

C o r o l l a r y 1. 0 (2 , 2, 2, 2) @ D(2, 2, 2, 2) = 0 ( 2 . 2 , 2- 2, 2, 2) is also a difference set in the residues (rood 2).

Summing up Theorems 1-4 and Corollary 1, we have the following T h e o r e m 6. D(2p, 2p, p, 2) exists for any prime p. Consider the Galois field GF(21+r) . The elements of the field can be expressed either

as powers x i of a primitive element x (i = 0, 1 , . - . , 2 l+r - 2) together witch the element zero, or as polynomials of degree r with coefficient from GF(2) , the field of residue classes (rood 2). To add two elements we use the polynomial form adding the coefficients (rood 2),

and to mult iply we use the power form remembering the relation x 21+~ = x. For example, if r = 2, we consider GF(23) , whose elements, by using the reduced

polynomial (minimum function) x 3 + x + 1, can be exhibited as

go = 0 - - - - 0 , g l ~ 1 ---- x °,

g5 = x2 + 1 = x 3,

g2 ~ x l ,

g6 ----- X2 Jr" ~ --~ ~6,

g3-~--xnul----X 5,

x7 ---- x 2 + x + 1 ---- x 4.

(2.1)

312 ACTA MATHEMATICAE APPLICATAE SINICA Vol . l l

We have ordered the" elements of the field in what may be called the lexicographic order, tha t is, if gl = azz 2 + a lx + a0 then the integer i is expressed as a2alao in the scale of numerate with radix 2. The same is done for the general case GF(2i+~). If

gl = ar x r -t- a r - l x r -1 Jr- - • • -1- az x -t- no, (2.2)

then i = a r a r - l ' " a l a o in the scale of numeration with radix 2. Consider the subclass M of the elements of GF(21+~) for which the coefficients of x ~

and higher powers of z are zero, when the element is expressed in the polynomial form. In our example the subclass M consists of the elements go, gz, g2, g3- In general M will consist of the first 2 r elements of GF(2 l+r) when they are arranged in the lexicographic order. Now we establish a correspondence between the elements of G F (2 z+') and the elements of M in the following manner. The element g~ of GF(2 z+r) given by (2.2) corresponds to the element

gj = a~ - lx ~-1 + - . - + a l z + ao (2.3)

of M, the coefficients of x r-1 and lower powers of x for gj being the same as the coefficients of the corresponding powers of z in g/. It is clear that a i is uniquely determined by g / a n d that j = i (rood 2r), 0 < j < 2 ~.

Conversely, to each gj of M there correspond 21 elements of GF(2Z+r), since if gj is given by (2.3) then for gi the coefficients a 2 - 1 , " " , ar are arbitrary, each taking 2 possible values. It should be noted that M is a direct factor module in GF(21+~). In our example the correspondence between the elements of GF(23) and M is given by

g4, go ~ go; g5, gl * gl; gs, g2 ~ g2; gT, g3 ~ g3. (2.4)

If we write down the multiplication table of GF(2 l+r) and then replace each element by the corresponding element in M, we get a 2 l+r × 21+~ matrix. If we take the difference of the corresponding elements in any two columns of the multiplication table, then every element of GF(21+~) occurs exactly once. Also if the elements g/, gl, of GF(21+~) correspond to the elements gj , gj , of M, then the element g~ -g~, of the field corresponds to the elementgj - g j , of M. This shows that in the matrix we have obtained each element of M occurs exactly )~ = 21 times among the differences of the corresponding elements of any two columns.

By the arguments above we have the following T h e o r e m 7 [3'1s]. The multiplication table of GF(21+~) gives a difference set D(21+~,

2 z+~, 2 l+r, 2) for r > 1. T h e o r e m 8 [3,1s]. There exists a difference set D(2- 2 ~, 2- 2 ~, 2 ~, 2) for r > 2. E x A m p l e 5. When r = 2, we have to write down the multiplication table of GF(23).

This can be done by using the identifications given in (2.1), remembering that x s = x. We thus get

"go go go go go g? go go" go gl g2 g3 g4 g5 gs g7 go g2 g4 g6 g3 gl g~ gs go gs g6 g5 g7 g4 gz g2 go g4 g3 g7 gs g2 g5 gl go gs gz g4 g2 g7 g3 gs go gs gv gz gs g3 g2 g4

-gO g7 g5 g2 gl gs g4 g3.

To obtain D ( 2 . 2 2 , 2- 22, 22, 2) we replace the g ' s in (2.5) replacements are go "-* 0, gl --* 1, g2 -"* 2 and g3 "-* 3.

(2.5)

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314 ACTA MATHEMATICAE APPLICATAE SINICA Vol.11

Summing up Theorems 8 and 9, we get the following T h e o r e m 10. Difference set D(2p", 2p", pr, 2) exists for any prime p and any positive

integer r. Combining Theorems 6 and 10 we get T h e o r e m 11. Difference set D(2p, 2p, p, 2) exists where p is prime or a prime power.

3 . C o n s t r u c t i o n o f O r t h o g o n a l A r r a y s

As shown in [3] a difference set D(Ap, Ap, p, 2) leads to an orthogonal array (Ap 2, Ap, p, 2) by replacing each element of the former by the corresponding column of the addition table of M, and then the latter is resolvable, i.e., the Ap 2 rows.can be divided into Ap sets of p each such that in any column of the array each set contains all the p elements exactly once. We can add one more column by putting the element z in any A sets, x E GF(p). Obviously then we get (Ap 2, Ap + 1, p, 2).

We now give another method for constructing orthogonal arrays. Let D = D(Alp, kl,p, 2) be a difference set and A = (A2p2,k2,p,2) an orthogonal array. Then it is easily seen that D $ A is an orthogonal array (A1A2p 3, klk2,p, 2) in the residues (rood p), which is also resolvable, i.e., the A1A2p a rows can be divided into Alp sets of A2p 2 each such that in any column of the array each set contains all the p elements exactly A2p times. We can add one more column by putting the element x in any A1 sets, x E GF(p). Obviously then we have (A1A:p 3, klk2 -4- 1,p, 2). We thus have the following

T h e o r e m 12. The existence of D(A1, kl,p, 2) and (A2p 2, k2,p, 2) implies the existence of (AIA2p 3, klk2 + 1,p, 2).

Hence, it follows from Theorem 12 that D(p,p,p, 2) and (p2,1 + p,p, 2) given by (1.1) lead to the orthogonal array

(V3,1 + p + p2,p, 2), (3.1)

where p is a prime or a prime power. Again from D(p,p,p, 2) and the orthogonal array (3.1) if follows that

(p4, 1 + p + p 2 +p3, p, 2) (3.2)

exists. Repeating thd procedure employed just now we finally obtain

(p'~, 1 + p + " " + p n - l , p, 2), (3.3)

where n > 2. Similarly, D(2p, 2p, p, 2) and (pn, 1 + p + . - . + pn--1, p, 2) give

[2p n+l, 1 A'- 2(p + p2 _{_... "4- pn), p, 2], (3.4)

where p is a prime or a prime power and n is any positive integer. Again from D(2p, 2p, p, 2) and the array (3.4) it follows that

[4p rt+2, 1 "~ 2p "4- 4(p 2 "4- p3 _~....~_ ~0n-t-1), p, 2] (3.5)

exists. Furthermore, D(2p, 2p, p, 2) and the array (3.5) give

[8p ~+~, 1 -{- 2p + 4p 2 + 8(p 3 + p4 +...p,~+2), p, 2]. (3.6)

Seiden [15] has constructed a difference set D(4 .3 , 4- 3, 3, 2) by trial and error. Hence we can construc~ two orthogonal arrays

[4- 3 "+I, 1 + 4(3 + 3 2 + - - " + 3"), 3, 2] (3.7)

No.3 A SURVEY OF ORTHOGONAL ARRAYS OF STRENGTH TWO 315

and [8.3 '~+2, 1 + 4 . 3 + 8(32 + 33 + . . . + 3'~+1), 3, 2] (3.8)

by using D(4 .3 , 4 .3 , 3, 2) and the array (3.3) for p = 3 and D(4 .3 , 4 .3 , 3, 2) and the array (3.4) for p = 3, respectively.

Note that the constraints of the orthogonai arrays (3.3) and (3.4) are the upper bonds for the maximum possible number (see [3]). Addelman and Kempthorne [11 and Shrikhande[ x61 have constructed the orthogonal array (3.4) by other procedures, although Shrikhande's method depending upon a difference set, p is not a prime power but a prime.

When p = 3 the constraints of the arrays (3.5) and (3.6) are less than those of the arrays (3.7) and (3.8) respectively.

For example, if p = 3 and n = 1 then we get from (3.5) an orthogonal array

(108, 43, 3, 2) (3.9)

and if n = 2 then we get from (3.7) an orthogonai array

(108,49,3,2). (3.10)

Clearly the constraints of the array (3.10) are larger than those of the array (3.9).

4. A Relationship Between ARBIB Designs and O r t h o g o n a l A r r a y s

A balanced incomplete block (BIB) design with parameters v, b, r, k, $ is an arrangement of v treatments into b blocks of k < v distinct treatments such that each treatment occurs in r blocks and any pair of treatments occurs in ~ blocks. A BIB design is called resolvable if the blocks can be separated into r sets each forming a complete replication of all the treatments. A resolvable BIB design is called an affine resolvable BIB (ARBIB) design if any two blocks of different sets have the same number of treatments in common. As shown in [2] the parameters of such a design are given in terms of two integers p > 2, q > 0 by

v = [(p - 1)q + 1]p 2, b = p(1 + p + p2q), )~ = pq + 1. (4.1)

Plackett and Burman [131 and Shrikhande[lS] have shown that the existence of an ARBIB design with parameters (4.1) implies the existence of an orthogonal array

[ ( ( p - 1)q+ 1)p 2, l+p+p2q, p, 2],

and vice versa. Bose [21 has already shown that the design (4.1) exists for

(4.2)

q = - - 1 , (4.3)

where p is a prime or a prime power. Conversely, if we take ( p - 1 )q + 1 for the index of the array (4.2) and solve ( p - 1 )q + 1 =

p'~ then we have the relation (4.3). Substituting (4.3) in (4.2) we have an array

[p,~+2, 1 + p + p 2 + . . . + p,~+l, p, 2]. (4.4)

Since there are D(p,p,p, 2) and (p,~+l, 1 + p + p2 + . . . + pn, p, 2) it follows from Theorem 12 that the array (4.4) is of orthogonal.

316 A C T A M A T H E M A T I C A E A P P L I C A T A E S I N I C A Vol . l l

On the o ther hand, Shr ikhande [18] has shown t h a t a necessary condi t ion for the existence of the A R B I B design (4.1) is t ha t

(i) (p - 1)q ÷ 1 should be a perfect square if p is odd and q is even, (ii) p [ (p - 1)q + 1] should be a perfect square if b o t h p and q are odd.

For example let p --- 3. I f q = 2 and 3 then (p - 1)q + 1 --- 5 and p [ (p - 1)q ÷ 1] = 21 respectively, b o t h being not perfect squares. Thus the designs

v = 4 5 , b = 6 6 , (4.5)

and v = 63, b = 93,

are impossible, and hence the corresponding arrays

= 10 (4.6)

(45, 22, 3, 2) (4.7)

and

are impossible

(63, 31, 3, 2) (4.8)

A c k n o w l e d g m e n t s . T he first au thor would like to express his hear ty thanks to President T s u t o m u Kake, O k a y a m a University of Science, and Director Sumiyasu Yamamoto , In terna- t ional Ins t i tu te for Na tu ra l Sciences, for their financial suppor t s and encouragements . The authors also wish to t hank Doc to r Yoshifumi Hyodo for his valuable discussions.

References

[1] Addelman, S. and O. Kempthorne. Some Main-effect Plans and Orthogonal Arrays of Strength Two. Ann. Math. Statist., 1961, 32: 1167-1176.

[2] Bose, R.C. A Note on the Resolvability of Balanced Incomplete Block Designs. Sankhya, 1942, 6: 105-110.

[3] Bose, R.C. and K.A. Bush. Orthogonal Arrays of Strength Two and Three. Ann. Math. Statist., 1952, 23: 508-524.

[4] Fujii, Y. Nonexistence of a Two Symbol Orthogonal Arrays of Strength 8b 11 Constraints and Index 6. TRU Math., 1988, 24(2): 153-165.

[5] Fajii, Y., T. Namikawa and S. Yamamoto. On Three-symbol Orthogonal Arrays. ISI, 1987, 46: 131-132.

[6] Fujii, Y., T. Nam~kawa and S. Yamamoto. Two-symbol Orthogonal Arrays of Strength ~ and t+3 Constraints. TRU Math., 1988, 24(1): 55-63.

[7] Fujii, Y., T. Namikawa and S. Yamamoto. Classification of Two-symbol Orthogonal Arrays of Strength t,t-{-3 Constraints and Index 4, H. SUT 3. Math., 1989, 25(2): 161-177.

[8] Jiang, S. A Simple Construction of 2p×2p Difference Scheme with Module p where p is an Arbitrary Odd Prime. Acta Math. Appl. Sinica~ 1979, 2:75-80 (in Chinese).

[9] Liu, Z.W. Construction of Difference Sets Which Generate Orthogonal Arrays (2p2,2p+l,p,2) where p is an Odd Prime. Acta Math. Appl. Sinica, 1977, 3:35-45 (in Chinese).

[10] Masuyama, M. On Difference Sets for Constructing Orthogonal Arrays of Index Two and of Strength Two. Rep. Star. AppÂ. Res., JUSE, 1957, 5: 27-34.

[11] Masuyama, M. Construction of Difference Sets for OA(2p2,2p+I,p,2), p Being an Odd Prime. ibib.,

1969, 16: 1-9.

No.3 A SURVEY OF ORTHOGONAL ARRAYS OF STRENGTH TWO 317

[12] Namlkawa, T., Y. Fujii and S. Yamamoto. Computational Study on the Classification of Two-symbol Orthogonal Arrays of Strength t and m = t + e Constraints for e <~ 3. SUT J. Math., 1989, 25(2): 179-195.

[13] Plackett, R.L. and J.P. Burman. The Design of Optimum Multifactorial Experiments. Biometrika, 1946, 33: 305-325.

[14] Rao, C.R. Factorial Experiments Derivable from Combinatorial Arrangements of Arrays. J. Roy. star. Soc. Suppl., 1947, 9: 128-139.

[15] Seiden, E. On the Problems of Construction of Orthogonal Arrays. Ann. Math. Statist., 1954, 25:

151-156. [16] Shrikhande, S.S. The Non-existence of Certain Affine Resolvable Balanced Incomplete Block Designs.

Canad. J. Math., 1953, 5: 413-420. [17] Shrikhande, S.S. Generalized Hadamard Matrices and Orthogonal Arrays of Strength Two. ibid.,

1964, 16: 736-7"40. [18] Xiang, K.F. The Difference Set Table for A=2. Acta. Math. AppL Sinlca, 1983, 6(2): 160-166 (in

Chinese).

[19] Xu, C,X. Construction of Orthogonal Arrays L2p, (p~+~-'~=lzP') with Odd Prime p. ibid., 1979, 2: 92-97 (in Chinese).

[20] Yamamoto, S., T. Namikawa and Y. Fujii. Classification of Two-symbol Orthogonal Arrays of Strength t,t-{-3 Constraints and Index 4. TRU Math., 1988, 24(2): 167-184.

[211 Yamamoto, S., Y. Fujii and M. Mitsuoka. Three-symbol Orthogonal Arrays of Strength 2 and Index 2 Having Maximal Constraints-computational Study. IINS Technical Report, No.6, International Institute for Natural Sciences, Kurashild, Japan, 1991.