incomplete orthogonal arrays and idempotent orthogonal arrays

Graphs and Combinatorics (1996) 12:253-266 Graphs and Combinatorics © Springcr-Verlag 1996

Incomplete Orthogonal Arrays and Idempotent Orthogonal Arrays

Francis Maurin

6, Rue Mizon, 75015, Paris, France

Abstract. First, we shall define idempotent orthogonal arrays and notice that idempotent orthogonal array of strength two are idempotent mutually orthogonal quasi-groups. Then, we shall state some properties of idempotent orthogonal arrays.

Next, we shall prove that, starting from an incomplete orthogonal array Te\ F based on E and F c E, from an orthogonal array Ta based on G = E - F and from an idempotent orthogonal array TH based on H, we are able to construct an incomplete orthogonal array T~vu~o×m~\v based on FU(G x H) and F.

Finally, we shall show the relationship between the construction which lead us to this result and the singular direct product of mutually orthogonal quasi-groups given by Sade [5].

1. Introduction

A.S. Hedayat and 1,1,1 Stufken studied in 1,2,1 certain fractional factorial designs which are known as incomplete orthogonal arrays in the writings concerned. They have described situations in which these designs can be of practical interest and studied their statistical properties, hence they could demonstrate the statistical usefulness of incomplete orthogonal arrays.

Nevertheless, since J.D. Horton studied incomplete orthogonal arrays of strength two in his paper I-31, few works have been published on the mathematical aspects and the construction of such designs. We began to be interested in these questions in 1,4,1.

Our purpose, here, is to show how, starting from an incomplete orthogonal array based on E and F c E, from an orthogonal array based on G = E - F and from an idempotent orthogonal array based on H, it is possible to construct an incomplete orthogonal array based on F U (G x H) and F.

We shall start by giving the definitions of incomplete orthogonal arrays and orthogonal arrays.

Definition 1. Let (ai4) be a (r x N) - matrix with entries chosen from the set E = {e 1, e: . . . . . es}. Let F be a k - subset of E and (P) the following property:

254 F. Maurin

the columns of any (t x N) - submatrix (t < r) of (a~j) are the elements of E' - F t each being repeated exactly 2 times.

A matrix (ai.j) satisfying (P) is an incomplete orthogonal array based on E and F, of strength t, of index 2 and of r constraints. We shall denote such an incomplete orthogonal array by the five integers (2, r,s, k, t). It is to be noticed that N = 2 ( s ' - k').

Definition 2. With the notations of definition 1, if F = ~, a matrix (a~.j) satisfying (P) is an orthogonal array based on E, of strength t, of index 2 and of r constraints. Let such an orthogonal array be denoted by the four integers (2, r, s, t). Notice that N = ),s t.

2. Idempotent Orthogonal Arrays

Definition. An orthogonal array To(l, r, s, t) is an idempotent orthogonal array if there exists a sequence of sub-orthogonal-arrays T~(I, r, s, t - i) of To and if T~ is a sub-orthogonal-array of T~_ 1, V i = 1, 2, . . . , t - 1.

R e m a r k 1. If To(1,r,s, t) is an idempotent orthogonal array, then T~ is also an idempotent orthogonal array V i = 1, 2, . . . , t - 1.

Remark 2. Since Tr-1 is an orthogonal array of strength 1, the set of its columns can be described by

(ei, ~l(e~),-.., ~(e:) . . . . . ~ ,_ l (e~) ) ,

V e~ e E the set on which To is based, and where n k is a permutation of E = {el, e2 . . . . . es},Vk = 1,2 . . . . , r - 1.

Hence, by applying the permutation lt~ 1 to the elements of the (k + 1)St-row of To, we obtain an idempotent orthogonal array T~ such that s columns of T~ are (e~, e~ . . . . . e~,..., e~), V i = 1, 2, . . . , s.

Then, considering the case "t = 2" it follows that idempotent orthogonal arrays come out as a generalization of idempotent mutually orthogonal quasi-groups.

R e m a r k 3. If there exists an orthogonal array O (1,r + t - 1,s,t), then it can be inferred that there exists an idempotent orthogonal array To(l, r, s, t).

Let O be the ( ( r + t - 1 ) x N ) - matrix (a=,B) ( ~ = 1 , 2 . . . . . r + t - l : t = 1, 2, . . . , N) and let T o be the (r x N) - submatrix of O, (at,a) (y = t, t + 1 . . . . . r + t - l : f l = 1,2 . . . . ,N).

Let Ti be the sub-matrix of To, (ar,~to) (Y = t, t + 1,... , r + t - 1) and where the 6(0 are such that al.~ 0 = a2 , ,~ (o --- . . . = a i j , ( 0 = e ~ E the set on which O is based.

In such a case, T~ is an orthogonal array ( 1 , r , s , t - i ) and T~ is a sub-orthogonal-array of Ti-1, V i = 1, 2 . . . . . t - 1. Thus indeed To is an idempotent orthogonal array (1, r, s, t). Remark 3 appears as a generalization of Mann's result: "from m mutually orthogonal quasi-groups, it is possible to construct ( m - 1) idempotent mutually orthogonal quasi-groups".

Remark 4. If there exists an idempotent orthogonal array To(l, r, s, t), then r < s.

Incomplete Orthogonal Arrays and Idempotent Orthogonal Arrays 255

Since an idempotent orthogonal array T,_2(1,r,s,2) does exist too, or, equivalently, (r - 2) idempotent mutually orthogonal quasi-groups do exist and it is a known result about idempotent mutually orthogonal quasi-groups that (r - 2) _ (s - 2).

Example 1. We know [1] that an orthogonal array T(1, 6, 4, 3) exists:

0 T 0

ro

1 T 1

ro

T 2

ro

T 3 ro

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 1 3 2 2 3 1 3 2 1 1 2 3 3 1 2 2 3 0 3 0 1 1 2 0 1 2 3 2 3 1 3 0 2 1 3 0 2 0 1 1 2 3 1 2 3 0 2 3 0 1 3 0 1 2 1 2 3 0 0 0 1 1 1 2 2 2 3 3 3

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 3 2 0 1 2 3 3 2 1 0 2 3 0 1 2 0 3 3 0 2 1 2 1 3 0 0 3 1 2 3 2 0 2 0 1 3 0 2 3 1 3 1 0 2 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 0 1 3 2 1 0 0 1 2 3 1 0 3 2 1 3 0 0 3 1 2 1 2 0 3 3 0 2 1 0 1 3 1 3 2 0 3 1 0 2 0 2 3 1 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 1 0 2 3 0 1 1 0 3 2 0 1 2 3 0 2 1 1 2 0 3 0 3 1 2 2 1 3 0 1 0 2 0 2 3 1 2 0 1 3 1 3 2 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3

As an application of remark 3, an idempotent orthogonal array To(I, 4, 4, 3) is obtained from T. The set of the 16 first columns of To constitutes TI and the set of the 4 first columns of To represents T2.

Example 2. The following orthogonal array To(l, 4, 4, 2) is an idempotent one:

0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 1 0 3 2 2 3 0 1 3 2 1 0 0 1 2 3 3 2 1 0 1 0 3 2 2 3 0 1 0 1 2 3 2 3 0 1 3 2 1 0 1 0 3 2

Yet, as there is no orthogonal array T(1, 6, 4, 2) since for an orthogonal array (1, r, 4, 2), r < 5. To cannot be constructed through applying remark 3.

3. Constructing Incomplete Orthogonal Arrays

We shall first prove two identities which we need in order to support the demon- stration of our main result.

256 F. Maur in

L e m m a 1.

(-I) Ci+w_iC ~, -0 Vr__l, l S a S r w=O

Le t us prove f i r s t that, f o r i < ~,

~ ( _ 1)wC~+w_lC=_,~ = ( _ 1),r( (r + i)! (or -- 1)(a -- 2 ) . . . (~ - i) w---o - ct + i)!cd it

(I)

I t is easy to see tha t (I) is t rue for i = 1. Le t us assume tha t (I) is t rue for i = k < ~ - 1 a n d let us p rove it for i = k + 1.

W e have then

k+l (r + k)! (~ - - 1)(~ - - 2 ) . . . ( = - - k)

w-----O

+ (- I) c;

= ( _ 1)k+ 1 (r + k)l (~ - 1) (a - 2 ) . . . (= - k)

(r + k + I -- a)!~! (k + 1)!

x [ - ( k + l ) ( r + k + l - ~ ) + r a ]

= ( _ 1)k+ 1 (r + k + 1)! (~ - 1)(ct - - 2 ) . . . (a - - k - - 1)

(r + k + 1 - ~,)!ctl (k + l)l

which, for i = k + 1,

is (I) indeed. Let us wri te (I) for i = a - 1:

• -1 ( - 1 ) " 1 ( ~ + : 1 ~ . ~ ) ' ( ~ - 1 ) , E ( - c , ' - - = - - •

w=o ( a - 1)l

_ _ = ( _ 1) ,_ 1 (r + ~ - 1)l (r - 1)!¢d '

hence

~=o - I ) ! ~ ! + (- - 1)=C~=+=_1C ° = 0 [ ]

L e m m a 2.

~ ( _ n w r , wr , -1 = 0 V ~ > r > l w~O

We f i r s t prove that, f o r i < r,

(a + i)! ( - n '~r~wr'-1 = ( - 1) i (II) .J ~, "-'w+,-1 (r - 1 i)!(~ + i - r)! i!~

I t is easy to see tha t (II) is t rue for i = 1. Let us assume tha t (II) is t rue for i = k < r - 1 and let us p rove it for i = k + 1. Thus , we have:


k+i (*( + k)! E w w r-1 (-1) c; c7+~_~

= ( - 1 ) k ( r - 1 k)!(~ + k r)!k!a w = O ~ - -

+ ( - 1)k +x r(~ + k)!

( r - k - l)!(a + k - r + 1)!(k + 1)!

= ( - l)k+ 1 (a + k)! ( r - - k - 1)!(a + k - r)!k!

-- + ( k + l ) ( g + k - r + l

= (-- 1) k+l (c( + k)!

( r - k - 1 ) ! ( e + k - r + l ) [ ( k + 1)!~

x [ - ( k + 1) ( : - r + k + 1) + r : ]

= ( - - 1) k+i (a + k)!

( r - k - 1)!(a + k - r + 1)!(k + 1)!a

x [ ( r - k - l)(a + k + I)]

= ( - i ) k+i (a + k + 1)!

( r - k - 2)! (e + k - r + 1)t(k + 1)!a

which, for i = k + 1,

is (II) indeed. Therefore, let us write (II) for i = r - 1:

,-1 r l ( c( + r - 1)! E w w r-I ~:o (- 1) C; C7+:_~ = (- I) - i~r ZZ i~.

hence

w w .-1 ( ~ + r - 1 ) ! , . ~-i w=o ( - 1 ) C; Cw+~-~ = ( - 1 ) '-1 ~ t . i ~ : 1~. + ( - 1 ) C;C',+~_I = 0 []

Theorem 1. Let E and H be two sets such that card E = s~ and card H = h, F a s2-subset o f E and G = E - F. I f there exist an incomplete orthogonal array based on E and F

Te\v(1, r + t, s 1, s2, t), an orthogonal array based on G

TG(1, r + 2t -- 2, sl -- s2, t) and an idempotent orthogonal array based on H

T.(1,r + t,h,t),

then there does exist an incomplete orthogonal array based on F U (G x H) and F

T~euta×n))\r((t -- 2)Ir'-2, r + t, S 2 + ( S 1 - - s2)h, s2, t).

We describe TE\F, Ta and T. by the set of their columns in the following manner:

258 F. Maurin

T~\ F = { [X] = (X, Et(X), E2(X) . . . . . Er(X)); V X • E' - F'}

r G = {[X'] = (GI(X') . . . . , G,-2(X') ,X' , G2,-t(X') . . . . . G2,-2+r(X')); V X ' • G'}

T. = { [Y] = (Y, H, (Y), Hz(Y, . . . . H,(Y)); V Y • H t}

where Ei(X) • E, Gj(X') • G and Hk(Y) • H.

Let T~, ( l , r + t , h , t - i) be the sequence of orthogonal sub-arrays of the idempotent orthogonal array TH such that TH, be an orthogonal sub-array of T~,_I (i = I, 2 . . . . . t - 1 and THo = Tx). Let us denote by H (° the set of columns of TH,_I- Tn, (i = 1,2,. . . , t - 1)and H ( ° = TH,_.

Existence of T~\r implies [4] the inequality: s t - s2 > rs2, for t _> 2, and hence there exists a family of disjoint s 2 - subsets f¢~ (i = 1, 2 . . . . . r) of G. Let us denote by p~ a one-to-one mapping of F on ~,.

The incomplete orthogonal array T~vo(o×m)\r which we want to construct will be described by the set of its columns:

= { [ Z ] = v z • ( F U ( G x H))' - F'}

where ¢~(Z) • F U (G x H).

Starting from the t-tuple Z, we wish to describe ~(Z), V i = 1 . . . . , r. For this purpose, some notations are needed:

X will be deduced from Z by leaving the components of Z belonging to F unchanged and by replacing the components of Z belonging to G × H by their projection on G; for example, i fZ = (.t"1,]'2 . . . . ,f~-k, (g~-k+t, h,-k+t), (g~-k+2, h,-k+2), . . . . (g,,ht)) for f~ • F, Yi = 1, 2 . . . . , t - k , g j • G , h i • H , Yj = t - k + 1, t - k + 2, . . . , t, then X = (ft,]'2 . . . . ,f~-k, g,-k+t, gt-k+2,... , gt).

Y will be deduce from Z as follows: if the k components of Z belonging to G x H are in the rows of rank it, i2 . . . . . ik, we shall, in TA,_,, look for the column [Y] determined by the following condition: the element of the ilth-row of [Y] is the projection on H of the element of the ilth-row of Z (l = 1, 2, . . . , k); for example, if Z is the one described, we shall, in T~,_k, look for the column [Y] the components of which are: ht in the lira-row, V I = t - k + 1, t - k + 2, . . . , t. Since TR,_k is an orthogonal array of strength k and index 1, [Y] is uniquely determined and Y too.

Now, let us consider a t-tuple Z the (t - k) components of which belong to F and the k components of which belong to G x H.

We shall examine the following cases:

1 °) k = 1. The column [Z] is set so that ~u(Z) = (E~(X), H~(Y)) (u = 1, 2, . . . , r), (in this case E,(X) e G, V u = 1, 2 . . . . . r) and we put the latter column [Z] rt-2(t - 2)! times in TcFuta~n,\r. Thus, we obtain C~s[ -t (sl - s2)h(t - 2)! / -2 columns.

2 °) k > l . a) if [Y] • n <°

Let us assume that E , (X) • F, V u • (al,a2,...,~k_~} and E , ( X ) • G, V u q~ {at, az . . . . . ak-~} for v = 1 or 2 or ... or k. More precisely, the column [Z] is set in assuming that ¢,(Z) = E,(X) V u • {~1, ~2, . . . , ~ -~} , ~,(Z) = (E,(X), H,(Y)) V u {at,a2,...,ak_~} and we put the so build column IZ] r~-2(t - 2)! times in


~) when e ~ 25,

=

~.(Z) = (G2,_z+.(X'), H . ( Y ) )

fl ) when e = 25,

We state e = {e~,e2, . . . ,Sk_Z_,} ~ {6~,6 2 . . . . ,6kin+o} = 6 where, i f D > 0, 0 < k - l - c _< inf(k - l, r)

or, s u p ( 0 , k - l - r ) _ < c < k - I

and where, if D _< 0, 0 _ k - l - c _< inf(k - l + D, r) or, with d = D + c,

s u p ( 0 , k - l - r + D) < d_< k - l + D.

Let z be a one- to-one m a p p i n g of e in R = {2t - 1, 2t . . . . . 2t - 2 + r}. Then the column [ Z ] is ob ta ined so that

V a = l, 2 , . . . , k - l - c

V 2t - 2 + u e R - z(e).

~,(Z) = (G2,_2+u(X'), Hu(Y)) V u = 1, 2 , . . . , r

The obta ined column [ Z ] is put in T~Fut ~ ×m~\r

t ~ + c 2 d! ~ ( - 1 ) ~ / + ' - 2 - ~ ( I - - W ) ! c ~ _ , times i fD > 0

~=o (d - w)!

1)~rl+,_2_~(l + c - - 2 - - w ) ! d Z(- times i fD < 0. ~=o (d - w)! -

T(ru~ ×m)\F. We then obta in

k t-k C~ s 2 (sz - s2)kh(t -- 2)! r t-2 columns.

b) i f [ e ] e H ~t-t+l) (l = 2 , 3 , . . . , k ) Let J = {Jr,J2 . . . . . Jr-k} be the set of the indices of rows of Z where are to be

found the elements f rom F. Let I = { i l , i 2 , . . . , ik} be the set of the indices of rows of Z where the elements f rom G x H are.

Let xj, be the element of the j~th-row of Z and (x~,, Yi,) be the element of the ivth-row of Z.

Let rc be a one- to-one m a p p i n g of J in {1, 2 . . . . , t - 2} and tr be a m a p p i n g of J in {1 ,2 , . . . , r} .

A co lumn [ X ' ] of TG is uniquely determined by the following conditions:

p~(i,)(xj~) is the ~z(j,)t~-component of [X ' ] , V # e {1, 2 . . . . . t - k}

x~is the( t - 2 + i , ) th-component of [X ' ] , Vv ~ {1 ,2 , . . . ,k}

Among the (t - 2) first rows of [ X ' ] there remain (k - 2) rows the indices of which are different from ~(j#), V/2 = 1, 2 , . . . , t - k.

Let us assume that, a m o n g the elements of these (k - 2) rows, there are k - l + D ( - ( k - l) _< D _< l - 2) ones which belong to

Q) ~ : x ~ ~q¢h, , Vog= 1 , 2 , . . . , k - l + D i=1

260 F. Maurin

Finally, as many columns [Z'] are set as there are possibilities in the choice of re, a and z, that is to say respectively:

( t - - 2)! rt_ k and r! (k - 2)! ' (r - k + l + c)!"

Let us cheek that, in every case, Z generates r~-a(t - 2)! columns [Z] :

I) D > O , k - l - r < O . Then, the number of columns [Z] put in Tcru(~xm)\e is:

(t - 2)[ ,-k k-z r! rk-z-c m ( k - 2 ) tr. c~o(r= - k + l + c ) [ ''k-'+D"' w=o ~" ( -1 )w

+ c - 2 - x r t+c-2-w(l ~ - ~ . I w)! C~,+w_ 1

and, if (c, w) are replaced by new indices (u = c - w, w):

(t - 2)! ,_k, k _ k-l + u -- 2) ~-i-~ -~ : - ~ . r t l + O)t X r'+'-2(l E t n . r ~ rk-t-.-w • ~ : o (D + u)! w:o ~ - " ~,+w-l'- ' .

k - l - u

but it results from lemma 1 that ~ t t - -Jn~r'w.~.+w-1 "~,:k-~-"-w = 0, Yu ~ k - I and w=O

= 1 when u = k - l, so that the number of columns [Z] is:

( t - 2 ) ! ,_~., l + k - l - 2 ) ! _ r , _ 2 ( t_2)! . ( k : ~1. r (~ - + D)!r*+k-'-2(l(D + k - l)!

2) D > O , k - l - r > O . Then, the number of columns [Z] put in T(eu( ~.n))\F is:

(t - 2)! r,_ ~ ~7~-' rt r-k-l-c aV (- 1)~rl+c-z-w (k 2)! ~ - , ( r k + l + c ) t

~,.Sk~|q.D#.4~.

( l + c - 2 - ×

or, if new indices (u = c - w, w) replace (c, w),

k-I + u - 2)! k-l-~ ( t - 2 ) [ r , - k ( k _ l + D ) [ ~, r,+~-a(l y" (_nwrw r~-,-~-w ~ - - 2)[ ~=k-,-, (D -Tt-u~ w=o *, --,+w-~,-,,

~-l-.-1 + u 2)! k.l-~ ( t - 2)! ,_~,. 1 + D)! ~ r '+"-=(l - ~ ( - 1 ) TM

+ ~ r i x - .=0 (D + u)[ w=,- , - , - .

I.., w t , - , k - l - u - w X ~'~r+w-1 ""r

F r o m lemma 1 it follows that

k - l - u w w ~ l ~ - l - u - w ( - 1 ) C~+~-I- , = 0 w h e n u # k - l , = l w h e n u = k - l

w=O

and, taking the new index w' = w - (k - I - r - u) instead of w,


k - l - u 2 ( l~wf~w ( ' , k - l - u - w 1)k-l-r-u l~W' f~r-1 C r w' ~ - ' J ~ , * ~ - l ' r = ( - ( - ' J " - ~ ' + ( k - i - . ) - I

w ~ k - l - r - - u w'=O

= 0

as a consequence of l emma 2, since k - l - u _> r > 1.

Eventually, we have indeed obta ined rt-2(t - 2)! columns [.2,].

3) D < O , k - l - r + D < O . Then, the n u m b e r of columns [Z ] put in T~Fu( ~ xn))\r is:

(t -- 2)!rt_ k k-t+D r! r k - z - , d ~ d 1) ~r~+c-2-w (r k + I + C)''-'R-'+D • ~, ( - (k - 2)! d=o - • w=o

x (1 + c - 2 - w)! C~+w_ 1 (d - w)t

or, if new indices (u = d - w, w) are taken instead of (d, w),

(t - 2)l t-k,, k-l+O l+u_D_2( l + U -- D - 2)[ k-I+O-. ( k : - ~ . r. t K - - I + D ) ! ~=o ~ r u! ~=oZ ( - -1) w

× C~._~C~, -~-~÷~-~. k - l + D - u

F r o m L e m m a 1 we know that ~ ( - I ) w C ~ w _ I C k- l -~+°-~ = 0 when u # w----0

k - l + D, = 1 when u = k - l + D, so that the number of columns I 'Z] obta ined is:

(t - 2)! rr_k( k _ l + D)!r l+k-z+D-°-2(l + k - l + D - D - 2)! r t -2( t 2) k (k 2)! ( k - l + D)!

4) D < _ O , k - l - r + D > O . The n u m b e r of columns [Z ] put in T(eu( ~ xm)\r is then:

d ( t _ 2 ) ! r , _ k k-Z+o rI (~k-t-~,/t ( 1)Wrl+~-2-~

X (r k + l + c)! "~k-,+O". X -- ( k - 2)! d=k-,-,+D -- w=0

X (l + C -- 2 -- W)! C~+w_ 1 (d - w)!

or, after taking new indices (u = d - w, w) instead of (d, w):

k-t+o + u -- D - 2)! k-Z+O-. ( t - - 2 ) ! ,-k,-- I + D)! ~, r z÷"-D-2(l u! ~ (--1)w ( k - - ~ . r t x - - = = k - l + D - , ~ = 0

C; + (t - 2)t ,-k,. k-I+O-,-1 X C~+~_1 k-t-~+O-w ~ r i x - - l + D)[ ~=o~ rt+"-D-2

( l + u - D - 2 ) [ k-z+D-. × U[ 2 i l~w~ w f 'k-l-u+D-w

~, - - "t.I " 'r+w--I ~'~r w = k - l + D - r - u

k - l + D - u

Consequent ly because of 1emma 1: ( - l)wC~+w_l C, ~-l-u+D-w = 0 when w = 0

262 F. Maurin

u ~ k - l + D , = l w h e n u = k - l + D a n d , t a k i n g n e w i n d e x w ' = w - ( k - l + D - r - u)being substituted for w,

k - l + D - u t nwr-w rk-t-.+D-w 1)k-t+D-,-. , - - , - ,+.-i"- ' , = ( - ~ ( - 1) w'

w = k - l + D - r - u w'=O

C;,7~k_z+~_,,~_l ¢~" = 0

as a consequence of lemma 2, since k - l + D - u > r > 1. Finally, we have indeed found rt-2(t - 2)l columns [ZJ. Thus, when [Y] ~ H tt-l+x~ (l = 2, 3 . . . . , k), k , -k C~ s 2 ($1 - $ 2 ) k ( h I - - h l - 1 ) r ' - 2 ( t - - 2)!

columns are obtained; therefore, when I runs over {2, 3 . . . . . k}, we obtain

k Cks~-k(S t - - s2)k(h I - - h ' - l ) r t - 2 ( t - 2)! = Cks~-k (S l - - s2)k(h k - - h ) r t ~ 2 ( t - - 2)!

1=2

columns. When rY] e H t°, we obtain CkS~-k(S l - - s 2 ) k h r t - 2 ( t - - 2)! columns. So all in all,

when k > 1, we have obtained k t -k C~ s 2 (St - s 2 ) k h ~ r t - 2 ( t - - 2)! columns, and, when k runs over {1, 2 . . . . , t}, we have constructed an array T~FUtG×m)\F of:

t C t l s ~ X ( s x - s 2 ) h r t - 2 ( t - 2)t + ~, Cks~ 'R( s t - - s 2 ) k h k r Z - 2 ( t - - 2)!

k--2

= ['(s2 + (st -- s2)h) t - s t2qr t -2 ( t - 2)!

columns which is just the number of columns of an incomplete orthogonal array

T(Fut~×lt ) ) \F(r t -2( t - - 2)!, r + t, S2 + (Sl - - s2)h , s2, t).

R e m a r k s . The method of construction explained above shows that:

(1) the columns of T~ such that, in the set of their (t - 2) first rows, (t - k) or more

than ( t - k) elements belong to U (~ are only combined with columns of i= t

TH,_~ - - H t°, Y k = 2 . . . . . t;

(2) the columns of T~ are never combined with those of Ht°; (3) the columns of H ¢tJ are always combined with those of TE\r; (4) let us consider Z in which exactly (t - k) components belong to F.

If [Y] ~ H tt~, only TE\ F and Tn are used for the constructing; then, among the r last rows of [Z], ( k - 1) or less than ( k - 1) components may belong to F. If [Y] e H tt-l) (l > 0), only T~ and TH are used for the constructing and then, among the r last rows of [Z'J, (k - 2) or less than (k - 2) components may belong to F.

The remains to prove that the thus constructed array T~rvt~m) is an incomplete orthogonal array ( r t - 2 ( t - 2)!, r + t, s2 + (s t - s2)h , s2, t).

Let T be (ai~, a~: . . . . . a,~) ~ (F U (G x H)) t, {i~, i[ . . . . . i~_~} c {il, i 2 . . . . . it} and {i '~,i~ . . . . . i~'} = ( i ~ , i ~ , . . . , i t } - { i ' t , i ' 2 , . . . , i ~ - k } .

Let us assume that a~j ~ F, Vj = 1, 2 . . . . , t - k and a~i, e G x H, V 1, 2, . . . , k. Then, let us have U ~ E ~ deduced from T by leaving a~j unchanged and by replacing


a~7 by its projection on G, and V ~ H k deduced from T by taking off aij and by replacing a¢ by its projection on H. We are looking for the replications of T in T(vuco ×m)\v when the nth-component of T is in the i~%row of Ttvut o ~m)\v which we denote by saying "T occupies place P in Ttvu( o ×m)J" Consequently, we have to prove that T occurs rt-2(t - 2)! times in place P in TtFu(a~m~\r.

I °) /f k = 1. Let us assume that card {i ' l , i '~,. . . , i;_l}N {1,2 . . . . . t} = t - k ' and " "' . , " 1. Then, as a conse- hence card {h,z2,.. z , _ l } l q { t + l , t + 2 . . . . . t + r } = k ' -

quence of Remark 4, U may only come from Tg\v. Furthermore, in Te\e, U occurs once and only once in place P. Moreover, H (')

is an orthogonal array of strength 1 and index 1; therefore, T occurs in place P in TtvutG×m)\rr~-2(t -- 2)! times.

2 °) / fk > 1. Let us assume that card {i~,i' 2 . . . . ,i~_k} ~ (1,2 . . . . . t} -- t -- k -- u and that card {i'~,i~ . . . . ,i~'} N {1,2,.. . , t} = k - v. Since the ( t - k) components of T belonging to F may only come from the elements of the (t - 2) first rows of To, as a consequence of Remark i), we have to look, in TB~_~, for the column I'V] which is determined by V in the following manner: if the cth-component of V is the projection of ajAc) on H, this component must be found in the if(c)th-row of TH~_~ (unicity of IV] results from the fact that Tn,_~ is an orthogonal array of strength k and index 1).

a) if [V] ~ H (°. Remarks 2) and 3) imply that only Te\e and T n have been used. U occurs once and only once in place P in Tg\ e. Since, in that case, construction 1 °) or construction 2°)a) has been used and since, in these two constructions each column [Z] is put rt-2(t - 2)! times in Tteu(o.m~\e, T occurs in place p in Tleuto×n))\~rt-2(t - 2)! times.

b) i f IV] E H ('-'+~) (I = 2 . . . . , k). Remark I) implies that only TG and T n have been used that is to say that T may only come from construction 2°)b). We know then that the (t - k) components of T belonging to F come from the (t - 2) first rows of To and that the k components of T belonging to G x H come from components of T o which are to be found in k fixed rows of To among the rows: t - 1, t, . . . , 2 t - 2 + r .

( t - 2 ) ! t - k Hence T allows us to determine ~ r columns of T o where it comes

from. Given one of these columns, let us suppose that, in the set of its (t - 2) first

rows, there are 2 components belonging to ~) (~i, in addition to the (t - k) ones w

i=1

which contributed to the determination of this column. Then T comes from a Z such that (t - k - u + j) components belong to F and (k + u - j ) components to

2t (u + v)t G x H (0 < j < 2) and that only from such Z and in - - ways.

(~. -- j)! j! (u + v --j)! Then T will appear in place P in T~ru(o ×m)\v, N times with

264 F. Maurin

(t - 2)! ,-k i.f(~,.+v)

N = -~ z - ~ r j=oE - -

2l (u + v)l i.f(x-j.,-.-~)x., (2 - j ) ! (2 -- j )! j ! (u + v -- j)! i=l"b i!(2 - - j -- i)!

X ( r - u - v ) !

( r - u - v - i)!

inf ( ;, - i - ~ k - ] - i - l) x (2 - i - j ) ! ( - 1)Wr ~-J-~-2.'~

w~O

( k ~ " - x 3: i - 2 - w ) !

In order to obtain N, 8 cases must be considered:

1) 2 < u + v , 2 - j < r - u - v a n d 2 - i - j < k - j - i - l :

~- j ~ - i - j ( t -2 ) ! r ,_ k Z Z E ( -1) " N = (k - 2)! j=o ~=o ~,=o

2!(u + v)t(r - u - v)trk-J-'-2-W(k - j - i - 2 - w)[

or, with z = i + j,

(t - 2)! t-k a x-w 1)Wrk_=_2_w(k -- z -- 2 -- w)I N = Z Z ( - ( 2 - z w)t

• w=O ==0

" C~+°C'_._o ~=0

( t - 2 ) ! ,-k x a-w t k - z - 2 - w ) ! - ~ ~ - . j , ~ , + ~ - l , ~ , r (2 z w)!

• w=O z=O -- --

or, with z' = z + w

• J. Z' 2 l r k - = ' - 2 ( k - - z ' - - 2 ) ! ~ w w ='-w ( t - 2 ) ! ,-k E • g : T ) . ~ .=o = ( - i ) c~+ ._~ c , ~ N <=o

yet, it follows from lemma 1 that

We then obtain:

z' E w w ~ t - w ( - 1 ) C;+w-lC~ 0 for z' ~ 0.

w=0

( t - 2 ) ! ,-k r k - 2 ( k - 2)! r ' - 2 ( t - 2)!. N --- ~ : ~ . i r 2! t~ =

2) for the 7 other cases, in the same manner, applying lemma 1 or 2, we obtain the same value r'-2(t - 2)! for N which concludes the proof of Theorem 1.

Example 3. Let us consider the abelian group E = {1, a, a 2, b, ab, a2b} a product of the two cyclic groups (1,a,a 2} and F = {1,b} with ab = ba. Then, Te\r = { (x, y, z, xyz), V (x, y, z) e E 3 - F ~ } is an incomplete orthogonal array (1, 4, 6, 2, 3). We know that (1) there exists an orthogonal array To(I, 5, 4, 3) and an idempotent orthogonal array Tn(1, 4,4, 3)(Example 1).

As a consequence of Theorem 1, there exists an incomplete orthogonal array (1,4,18,2,3).

Incomplete Orthogonal Arrays and Idempotent Orthogonal Arrays

4. Constructing Orthogonal Arrays

265

Theorem 2. Let E and H be two sets such that card E = s 1 and card H = h, F be a s2-subset of E and G = E - F. I f there exist an orthogonal array TE(1, r + t, S 1, t) with an orthogonal sub-array Tr(1, r + t, s2, t), an orthogonal array (To(l, r + 2t - 2, sl - s2, t) and an idempotent orthogonal array Tn(1, r + t, h, t), then there exists an orthogonal array Trut~ ×m((t - 2)! r t-2, r + t, s 2 + (sl - - s 2 ) h , t) which has Tr as an orthogonal sub-array.

Indeed, existence of TE and Tr implies that of an incomplete orthogonal array TE\~(1, r + t, sl, s2, t).

Theorem 1 enables us to build an incomplete orthogonal array T~ru~6 ×m)\r((t - 2)! r t-: , r + t, s2 + (sl - s2)h, s2, t) and adding (t - 2)! r '-2 replications of Tr to Ttru~o×m)\e allows us to obtain Tvu(o×m.

Example 4. Let us come back to example 3 with

T e = {(x,y,z , xyz), V(x ,y ,z ) ~ E 3 }

TF = {(x,y,z , xyz), V(x ,y , z ) ~ F 3 }

an orthogonal sub-array of Te. By applying theorem 2, we obtain an orthogonal array Truth×m(1,4,18,3)

which has Te as an orthogonal sub-array. In the case t = 2, theorem 2 writen in terms of mutually orthogonal quasi-

groups gives the following corollary:

Corollary 2 (Sade). I f there exist r mutually orthogonal quasi-groups based on E, with r mutually orthogonal sub-quasi-groups based on F c E, r mutually orthogonal quasi-groups based on G = E - F and r idempotent mutually orthogonal quasi- groups based on H, there do exist r mutually orthogonal quasi-groups based on F U (G x H) with r mutually orthogonal sub-quasi-groups based on F.

We have thus succeeded in obtaining, the singular direct product of Sade (5) as a particular case of Theorem 2.

References

1. Busch, K.-A.: Orthogonal arrays of index unity. Ann. Math. Statist. 23, 426-434 (1952) 2. Hedayat, A.S., Stutken, J.: Fractional factorial designs in the form of incomplete

orthogonal arrays Statistical designs: theory and practice, proc. of the conference in honor of W.T. Federer (1987)

3. Horton, J.D.: Sublatin squares and incomplete orthogonal arrays. J. Comb. Theory Ser. A 16, 23-33 (1974)

266 F. Maurin

4. Maurin, F.: On incomplete orthogonal arrays. J. Comb. Theory Ser. A 40, 183-185 (1985)

5. Sade, A.: Produit direct singulier de quasi-groupes orthogonaux et anti-ab61iens. Ann. Soc. Sci. Bruxelles, s6r. I, 74, 91-99 (1960)

Received: October 17, 1994 Revised: October 20, 1995

incomplete orthogonal arrays and idempotent orthogonal arrays

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