isid/ms/26/2 Fbruary, 26 http://www.isid.ac.in/ statmath/indx.php?modul=prprint Construction of asymmtric orthogonal arrays of strngth thr via a rplacmnt mthod Tian-fang Zhang, Qiaoling Dng and Alok Dy Indian Statistical Institut, Dlhi Cntr 7, SJSS Marg, Nw Dlhi 6, India
Construction of asymmtric orthogonal arrays of strngth thr via a rplacmnt mthod Tian-fang Zhang, Qiaoling Dng Collg of Mathmatical and Informational Scincs, Jiangxi Normal Univrsity, Nanchang 3322, China Alok Dy Indian Statistical Institut,Nw Dlhi 6, India Abstract: A rplacmnt procdur to construct orthogonal arrays of strngth thr was proposd by Sun, Das and Dy (2). This mthod was latr xtndd by Sun and Dy (23). In this papr, w furthr xplor th rplacmnt procdur to obtain som nw familis of orthogonal arrays of strngth thr. MSC: 62K5 Kywords: Galois fild; Orthogonal arrays; Rplacmnt mthod; Strngth thr. Introduction and Prliminaris Asymmtric orthogonal arrays introducd by Rao (973) hav rcivd wid attntion in rcnt yars. Such arrays ar usful in xprimntal dsigns as univrsally optimal fractional factorial plans and thir us in industrial xprimnts for quality improvmnt has also bn widsprad. Th construction of asymmtric orthogonal arrays of strngth two hav bn studid xtnsivly, and on may rfr to Hdayat, Sloan and Stufkn (999) for a comprhnsiv account of ths. Mor rcnt work on orthogonal arrays of
strngth two includ thos by Sun and Kuhfld (25) and Chn, Ji and Li (24). Mthods of constructing asymmtric orthogonal arrays of strngth gratr than two hav not bn studid as xtnsivly as thos of strngth two. Som of ths mthods can b found.g., in Hdayat t al. (999), Dy and Mukrj (999), Sun, Das and Dy (2), Sun and Dy (23), Jiang and Yin (23) and Zhang, Zong and Dy (26). Sun t al. (2) proposd a rplacmnt procdur for rplacing a column with 2 k -symbols in an orthogonal array of strngth thr by svral 2-symbol columns to obtain nw familis of tight asymmtric orthogonal arrays of strngth thr. Th rplacmnt procdur of Sun t al. (2) was xtndd by Sun and Dy (23). In this papr, w furthr xplor th rplacmnt procdur to obtain svral nw familis of orthogonal arrays of strngth thr. Som of th constructd arrays ar tight. Rcall that an orthogonal array OA(N, n, m m n, g) of strngth g is an N n matrix with lmnts in th ith ( i n) column from a finit st of m i ( 2) distinct lmnts, such that in vry N g subarray, all possibl combinations of lmnts appar qually oftn as rows. Whn m = m 2 = = m n = m, say, w hav a symmtric orthogonal array, dnotd by OA(N, n, m, g); othrwis th array calld asymmtric. It is wll-known that in an OA(N, n, m m n, 3) of strngth thr, } { n N + (m i ) + (m n ) (m i ) (m ), () i= i= m i. Arrays of strngth thr attaining th abov bound whr m = max i n ar calld tight. Throughout, following th trminology in factorial xprimnts, w call th columns of an OA(N, n, m m n, g) factors, and dnot ths factors by L, L 2,, L n. Lt GF (s) dnot a Galois fild of ordr s. W can thn writ th lmnts of GF (s) as {,, w, w 2,, w s 2 }, whr, ar th idntity lmnts of GF (s) with rspct to th oprations of addition and multiplication, rspctivly and w is a primitiv lmnt of GF (s). Throughout, for a matrix A, A T dnots its transpos. W shall nd th following rsult. Lmma. Lt α, β b two lmnts of GF (s) such that α 2 = β 2. Thn (i) α = β, if s is vn, (ii) ithr α = β or α = β, if s is odd. If α, α,, α s ar th lmnts of GF (s), thn th st S = {α 2, α 2,, α 2 s } contains all th lmnts of GF (s), if s is vn. If s is odd, thn 2
th lmnts of th st S ar and (s )/2 non-zro lmnts of GF (s), ach non-zro lmnt apparing twic in S. For th factor L i ( i n), dfin u i columns, ach of ordr t ovr GF (s), say d i, d i2,, d iui. Thus, for th n factors, w hav in all n i= u i columns. Also, lt b a s t t matrix whos rows ar all possibl t-tupls ovr GF (s). Sun t al. (2) provd th following rsult. Thorm (Sun t al., 2). Considr a t n i= u i matrix H = A, A 2,, A n, A i = d i, d i2,, d iui, i n, such that for any choic of g matrics A i, A i2,, A ig from A, A 2,, A n, th t g j= u ij matrix A i, A i2,, A ig has full column rank ovr GF (s). Thn an OA(s t, n, (s u ) (s u 2 ) (s un ), g) can b constructd. 2 A Rplacmnt Mthod Lt s = p k, whr p is a prim and k( 2) is an intgr. W start with an asymmtric orthogonal array OA(s m, n, (s r ) s n, 3) (not that whn r =, th starting orthogonal array is a symmtric orthogonal array OA(s m, n, s, 3)). Following a rplacmnt procdur, an asymmtric orthogonal array OA(ps m, t + u +, (ps r ) s t p u, 3) can b constructd, whr t, u ar intgrs. Our mthod of construction involvs th following stps. Stp : Construct an asymmtric orthogonal array OA(s m, n, (s r ) s n, 3). This array can b constructd using Thorm by slcting n matrics ovr GF (s), namly A, A 2,, A n, such that for any choic of 3 distinct matrics A i, A j, A k, i, j, k {, 2,, n}, th matrix A i, A j, A k has full column rank. For xampl, an array OA(s 4, s+2, (s 2 ) s s+, 3) (s = 2 k ) can b constructd T by taking ths matrics as A =, A 2 =,,, T, A 3 =,,, T, A 4+i =, w 2i,, w i T, i s 2, whr w is a primitiv lmnt of GF (2 k ). Stp 2: Not that for obtaining th OA(s m, n, (s r ) s n, 3), w us th lmnts of GF (s), s = p k. Howvr, w find it mor convnint to us lmnts of GF (p) than thos of GF (p k ). In ordr to rplac th lmnts of GF (p k ) by thos of GF (p), w nd a matrix rprsntation of th lmnts of GF (p k ), whr th ntris of ths matrics ar th lmnts of GF (p). Lt th irrducibl polynomial of GF (p k ) b w k + α k w k + + α w + α, whr w is a primitiv lmnt of GF (p k ), α j GF (p), j k. Thn 3
th companion matrix of th irrducibl polynomial is α α W = α 2.. α k A typical lmnt w i of GF (p k ) corrsponds to a k k matrix W i with ntris from GF (p), whr is rprsntd by a null matrix of ordr k and is rprsntd by idntity matrix of ordr k. Rplacing th lmnts of GF (p k ) in A, A 2,, A n by thos of GF (p), w gt matrics A, A 2,, A n, whr A is an mk rk matrix with lmnts from GF (p), and A j (2 j n) ar of ordr mk k. Nxt, dfin th following matrics: T T P = A, P i = A, i = 2,, n, (2) i whr is a null vctor. Lt L = P, P 2,, P n and b a p mk+ (mk + ) matrix with rows as all possibl (mk + )-tupls ovr GF (p). Tak th product L and rplac th p rk+ = ps r distinct combinations undr th (rk + ) columns of P by ps r distinct lvls of L as wll as th p k = s distinct combinations undr th k columns of P j by s distinct lvls of th factor L j for ach j, 2 j n. Th array OA(ps m, n, (ps r ) s n, 3) can now b constructd via Thorm. Stp 3: Finally, w giv a mthod to construct an orthogonal array of typ OA(ps m, t + u +, (ps r ) s t p u, 3). To obtain this family of orthogonal arrays, w rplac a p k -lvl column by svral p-lvl columns in th array OA(ps m, n, (ps r ) s n, 3). Th rplacmnt procdur is as follows. Lt 2 b a matrix of ordr k h. Considr th matrics P i (i {2,, n}) dfind abov. Lt M i = P i 2 = T A i 2, 2 i n, (3) thn th lmnts of th first row of M i ar all zros. Substituting th first row of all zros by a row of all ons, th matrics Q i = A (4) i 2 4
ar obtaind. This rplacmnt procdur dos not disturb th orthogonality and can b usd for ach factor P i, 2 i n. Thn th array OA(ps m, t + u +, (ps r ) s t p u, 3), t n can b constructd by choosing th matrix H = P, P 2,, P t+, Q t+2,, Q n, whr Q i = q i, q i2,, q i,h and q ij is th column of th factor P t++(i t 2)h+j with p symbols, t + 2 i n, j h. Thus, following th abov stps, an orthogonal array OA(ps m, t + u +, (ps r ) s t p u, 3) can b constructd. Sinc 2 dpnds on whthr s is vn or odd, w dal with ths two cas sparatly. In th nxt sction, w construct svral familis of asymmtric orthogonal arrays of strngth thr. 3 Construction of Orthogonal Arrays Whn s is a powr of two, svral tight orthogonal arrays of strngth thr wr obtaind by Sun t al. (2) by invoking Thorm and also via a rplacmnt procdur. In this sction, w first obtain two nw familis of orthogonal arrays of strngth thr, whn s is a powr or of two. To bgin with, w hav th following rsults. Lmma 2. Lt b a k (2 k ) matrix whos columns ar all possibl k- tupls ovr GF (2), xcluding th null column and G b dfind as G =, whr is a row of all ons of ordr 2 k. Thn any thr columns of G ar linarly indpndnt. Proof. Sinc ach of th matrics of th following thr typs has full column rank, choosing any thr columns g, g 2, g 3 of G, in th matrix g, g 2, g 3 thr xits a 3 3 subarray with on of th abov thr typs. This can b don through lmntary row and column oprations. So any thr columns of G ar linarly indpndnt. Thorm 2. Suppos A, A 2,, A n ar matrics ovr GF (2 k ) with lmnt at th sam position, and any thr of thm ar linarly indpndnt. If th matrix F = P, P 2,, P t, Q t+,, Q n is obtaind by th rplacmnt 5
mthod dscribd abov for s vn, thn any thr factors of F ar linarly indpndnt. A proof of this Thorm is givn in Appndix. Using Thorm 2, w hav th following familis of tight arrays. Thorm 3. If s is a powr of two, thn a tight array OA(2s 4, s 2 (s 2)t, (2s 2 ) s t 2 (s+ t)(s ), 3) can b constructd for t s +. Proof. First, w construct an array OA(s 4, s+2, (s 2 ) s s+, 3). According to Sun t al. (2), th matrics corrsponding to th factors ar as follows: A = A 2 = A 3 = A 4+i = w 2i w i i s 2, whr w is a primitiv lmnt of GF (2 k ). Thn th array OA(s 4, s+2, (s 2 ) s s+, 3) can b constructd by Thorm. For k 3, ach matrix A k has lmnt at th sam position, and by Thorm 2, w know that any thr factors of F = P 3, P 4,, P t, Q t+,, Q s+2 ar linarly indpndnt. y th rplacmnt stps, P 2 is obtaind via th rplacmnt of A 2. Nxt, w prov th orthogonality of P 2 and F, for which w nd to considr six cass: P 2, P a, P b, P 2, P a, Q b, P 2, Q a, Q b, Q 2, P a, P b, Q 2, P a, Q b, Q 2, Q a, Q b, a, b 3. To sav spac, w provid a proof of only th cas Q 2, Q a, Q b ; th othr cass can b handld in a similar fashion. Rplacing th s-symbol column in P 2, P a, P b by 2 k columns with 2 symbols ach, w gt Q 2 =, Q a = W 2i W i, Q b = W 2j W j whr is a (2 k ) vctor of all ons, is a null matrix of ordr k (2 k ) and, is a matrix of ordr k (2 k ) whos columns ar all possibl k-tupls ovr GF (2), xcluding th null column. Choosing any thr columns q 2, q a, q b of Q 2, Q a, Q b, th matrix q 2, q a, q b xists a 3 3 subarray. 6
with on of th following typs: α whr α is any lmnt of GF (2). Hnc an array OA(2s 4, s 2 (s 2)t, (2s 2 ) s t 2 (s+ t)(s ), 3) can b constructd for t s +. Th tightnss of th array follows from (). Thorm 4. If s is a powr of two, thn a tight array OA(2s 5, s 3 (s 2)t, (2s 2 ) s t 2 (s2 +s+ t)(s ), 3) can b constructd for t s 2 + s +. Proof. First, construct an array OA(s 5, s 2 +s+2, (s 2 ) s s2 +s+, 3). Following Sun t l. (2), dfin th matrics corrsponding to th factors as: A = T, A 2 = T, A 3,, A s+2 ar of th form, α 2,,, α, α GF (s), and A s+3,, A s 2 +s+2 ar of th form β 2, γ 2,, β, γ, β, γ GF (s). Thn th array OA(s 5, s 2 + s + 2, (s 2 ) s s2 +s+, 3) can b constructd via Thorm. For 3 k s + 2, ach matrix A k has lmnt at th sam position, by Thorm 2, any thr factors of F = P 3, P 4,, P t, Q t +,, Q s+2 ar linarly indpndnt. In a similar way, any thr factors of F 2 = P s+3, P s+4,, P t2, Q t2 +,, Q s 2 +s+2 ar linarly indpndnt. y th rplacmnt stps, P 2 is obtaind from A 2. Th orthogonality of P 2, F and F 2 can b handld as in th cas of Thorm 3. Hnc an array OA(2s 5, s 3 (s 2)t, (2s 2 ) s t 2 (s2 +s+ t)(s ), 3) can b constructd for t s 2 + s +. Th tightnss of th array follows from (). W now considr th cas whn s is an odd prim powr. W confin to th cas whn s is a powr of 3 as, arrays for this cas ar not too larg. For othr valus of s bing an odd prim powr, mthods similar to s = 3 k can b mployd, but thn th siz of th arrays will b quit larg. Whn s is a powr of thr, th lmnts of GF (s) can b writtn as { w s 3 2,, w, w,, w, w,, w s 3 2 }. An orthogonal array OA(s m, n, (s r ) s n, 3) (s = 3 k, k ( 2) is an intgr) can b constructd by slcting n matrics ovr GF (3 k ), namly A, A 2,, A n such that ths matrics satisfy th rank condition of Thorm. Hr also, for constructing an OA(s m, n, (s r ) s n, 3), it is 7
mor convnint to us lmnts of GF (3) than thos of GF (3 k ). So an array OA(3s m, n, (3s r ) s n, 3) can b constructd by th rplacmnt stps of Sction 2. To obtain an orthogonal array of th typ OA(3s m, t + u +, (3s r ) s t 3 u, 3), w rplac a 3 k -symbol column by svral 3-symbol columns. Similar to th cas whn s is vn, w start by rplacing a column with 3 k symbols by 3 k columns with all possibl k-tupls ovr GF (3), xcluding th null column. W rplac a 3 k -symbol column by 2k columns ach with 3 symbols, without disturbing th orthogonality. Lt 3 = I k, 2I k, I k is idntity T matrix of ordr k. From (2), (3) and (4), w gt M i = P i 3 = T A i 2A, i Q i =, 2 i n, whr, as bfor, is a null vctor and is a A i 2A i row of all ons. It can b sn that th factors L i (2 i n) corrsponding to 3 k symbols can b rplacd by 2k factors with 3 symbols ach, dnotd by th matrix Q i, without disturbing th rank condition. Thn th array OA(3s m, (n t)2k+t+, (3s r ) s t 3 2k(n t), 3) ( t n ) can b constructd by choosing th matrix H = P, P 2,, P t+, Q t+2,, Q n, whr Q i = q i, q i2,, q (i,2k) and q ij is th column of th factor L t++(i t 2)k+j with 3 symbols, t + 2 i n, j 2k. Hr is an xampl to illustrat th abov stps of construction. Exampl. Suppos s = 9 = 3 2 so that k = 2. W start from constructing a symmtric orthogonal array OA(9 3,, 9, 3). Dfin th following matrics corrspond to th factors: A =,, T, A 2 =,, T, A 3 =, w, w 2 T, A 4 =, w 2, w 4 T, A 5 =, w 3, w 6 T, A 6 =, w 4, w T, A 7 =, w 5, w 2 T, A 8 =, w 6, w 4 T, A 9 =, w 7, w 6 T, A =,, T, whr w is a primitiv lmnt of GF (3 2 ). Thorm can now b usd to construct an orthogonal array OA(9 3,, 9, 3). An irrducibl polynomial of GF (3 2 ) is takn as w 2 + w + 2. Thn th companion matrix is W = 2 and th lmnts of GF (3 2 ) can b rprsntd by 2 2 matrics,, w 2,, w 2 2 2 2, w 3 2 2 2, 8
w 4 2 2, w 5 2 2, w 6 2, w 7 Rplacing th lmnts of GF (3 2 ) in A, A 2,, A by th abov matrics, w gt matrics A i ( i ) with th lmnts from GF (3) as A = T,, A = From th matrics A i ( i ), w gt th matrics P i ( i ), whr T T P = A, P j = A, 2 j. j According to th prvious analysis and using th matrics P i ( i ), th array OA(3 9 3,, 27 9 9, 3) can b constructd. W nxt illustrat th rplacmnt procdur for rplacing a 9-symbol column by 4 columns, ach with 3-symbols. For xampl, th first 9-symbol column dnotd by P 2, can b writtn as 2 2 P 2 = 2 = Q 2. 2 2 2 Th othr 9-symbol columns can b handld in a similar way. Hnc th array OA(3 9 3, 37 3t, 27 9 t 3 36 4t, 3) ( t 9) can b constructd by choosing th matrix H = P, P 2,, P t+, Q t+2,, Q, whr th matrix Q i has four columns, ach having 3 symbols. For obtaining mor orthogonal arrays, w nd th following rsults. Lmma 3. Lt = I k, 2I k, whr I k is an idntity matrix of ordr k. Dfin th matrix G as G =, whr is a row of all ons of ordr 2k. Thn any thr columns of G ar linarly indpndnt. Proof. Choos any thr columns g, g 2, g 3 of G. Suppos th thr columns g, g 2, g 3 ar from or, thn clarly, th matrix g I k 2I, g 2, g 3 has k. T. 9
full column rank. Suppos th thr columns g, g 2, g 3 ar from and I k, rspctivly. Thn th matrix g 2I, g 2, g 3 xists a 3 3 subarray with k on of th following typs by conducting row and column transformation: 2 2 2 Hnc any thr columns of G ar linarly indpndnt. 2 Thorm 5. Suppos A, A 2,, A n ar matrics with lmnt at th sam position ovr GF (3 k ), and any thr of thm ar linarly indpndnt. If th matrix F = P, P 2,, P t, Q t+,, Q n is obtaind by th rplacmnt mthod dscribd arlir, thn any thr factors of F ar linarly indpndnt. A proof of this Thorm is givn in th Appndix. y Thorm 5, w hav th following rsults. Thorm 6. If s is a powr of thr, thn an array OA(3s 4, 2sk (2k )t + 2k +, (3s 2 ) s t 2 2k(s+ t), 3) can b constructd for t s +. Proof. First, construct an array OA(s 4, s + 2, (s 2 ) s s+, 3). Following Sun t al. (2), th matrics corrsponding to th factors ar A = A 2 = A 3 = A 5+2i = w 2i w i A 4+2i =. w 2i w i i s 3 2, whr w is a primitiv lmnt of GF (3k ). Thn th array OA(s 4, s + 2, (s 2 ) s s+, 3) can b constructd by Thorm. For k 3, ach matrix A k has lmnt at th sam position, and by Thorm 5, w know that any thr factors of F = P 3, P 4,, P t, Q t+,, Q s+2 ar linarly indpndnt. y th rplacmnt stps, P 2 is obtaind from A 2. Lt N = {3, 4, 6, 8,, s + }, N 2 = {5, 7, 9,, s + 2}. Nxt, w prov th
orthogonality of P 2 and F, which has 8 cass. To sav spac, w prov only th cas Q 2, Q a, Q b, a N, b N 2. Rplacing th s-symbol column in P 2, P a, P b by 2k columns with 3-symbols ach, w gt Q 2 =, Q a = W 2i W i, Q b = W 2j W j whr is a row of ons of ordr 2k, is a null matrix of ordr k 2k and = I k, 2I k. Choosing thr columns q 2, q a, q b of Q 2, Q a, Q b, in th matrix q 2, q a, q b thr xists a 3 3 subarray of on of th following typs: α 2 α 2 α, 2 2 α whr α is any lmnt of GF (3). Othr cass can b handld in a similar fashion. Hnc an array OA(3s 4, 2sk (2k )t + 2k +, (3s 2 ) s t 2 2k(s+ t), 3) can b constructd for s = 3 k, t s +. Thorm 7. If s is a powr of thr, thn an array OA(3s 5, 2(s + )ks (2k )t +, (3s 2 ) s t 3 2k(s2 +s t), 3) can b constructd for t s 2 + s. Proof. First, construct an OA(s 5, s 2 + s +, (s 2 ) s s2 +s, 3). Following Zhang t al. (26), dfin th matrics corrsponding to th factors: T A =, A 2 = 2 T, A 3,, A s+ of th form, α 2,,, α, α GF (s), α, and A s+2,, A s 2 +s+ of th form β 2, γ 2,, β, γ, β, γ GF (s). Thn an array OA(s 5, s 2 + s +, (s 2 ) s s2 +s, 3) can b constructd by Thorm. For 3 k s +, ach matrix A k has lmnt at th sam position, and by Thorm 5, any thr factors of F = P 3, P 4,, P t, Q t +,, Q s+ ar linarly indpndnt. In a similar way, any thr factors of F 2 = P s+2, P s+3,, P t2, Q t2 +,, Q s 2 +s+ ar linarly indpndnt. y th rplacmnt stps, P 2 is obtaind from A 2. Th orthogonality of P 2, F and F 2 can b handld in a similar way as in Thorm 6. Hnc th array OA(3s 5, 2(s + )ks (2k )t +, (3s 2 ) s t 3 2k(s2 +s t), 3) can b constructd for t s 2 + s.
Appndix Proof of Thorm 2. Without loss of gnrality, st A i =, i n. y th rplacmnt mthod, w obtain A Ik D i T i = Di, P i = I k Q i = Di Di whr I k is idntity matrix of ordr k, is a null vctor of ordr k, is a row of all ons of ordr 2 k, is a matrix of ordr k (2 k ) whos columns ar all possibl k-tupls ovr GF (2), xcluding th null column. Choosing any thr factors of F hav th following cass: Cas : Lt th thr columns f, f 2, f 3 b from Q i. This cas thn follows from Lmma 2. Cas 2 : Lt th thr columns f, f 2, f 3 b from Q i, Q j, Q k, i, j, k ar all distinct. f is th i -th column of Q i, f 2 is th j -th column of Q j, f 3 is th k -th column of Q k. (a) Suppos i, j, k ar all distinct. This cas follows from Lmma 2. (b) Suppos i = j k, w can choos a row in and D i, rspctivly. Thn in th matrix f, f 2, f 3 thr xists a 3 3 subarray is on of th following typs: α whr α is any lmnt of GF (2). α (c) Suppos i = j = k. W can choos two rows in D i. Thn in th matrix f, f 2, f 3 thr xists 3 3 subarray with th sam typs as in cas (b). Cas 3 : Lt two columns f, f 2 b from Q i, f 3 is from Q j, i j. f, f 2 ar th i -th, j -th columns of Q i, f 3 is th k -th column of Q j. Suppos i j = k. Thn this cas is th sam as Cas (2b). Suppos i j k, thn this cas is sam as Cas (2a). 2
Cas 4 : Choosing th matrix P i, two columns f, f 2 ar from Q j. In th matrix P i, f, f 2 thr xists a (k + 2) (k + 2) subarray of on of th following typs: T I k α T I k α whr α is any lmnt of GF (2). Thn this matrix P i, f, f 2 has rank k + 2. Cas 5 : Choosing th matrics P i, P j, and a column f is from Q k. matrix T T P i, P j, f = I k I k b Di Dj Dkb whr b is a column of, and this matrix has rank 2k +. Th Hnc any thr factors of F ar linarly indpndnt. Proof of Thorm 5 Without loss of gnrality, st A i =, i n. y th rplacmnt mthod for s odd, w obtain Q i = I k 2I k, whr is a row D i of all ons with ordr k, I k is idntity matrix of ordr k. Choosing any thr factors of F, w hav th following cass: Cas : Lt thr columns f, f 2, f 3 b from Q i. This cas follows from Lmma 3. Cas 2 : Lt thr columns f, f 2, f 3 b from Q i, Q j, Q k, i, j, k ar all distinct. f is th i -th column of Q i, f 2 is th j -th column of Q j, f 3 is th k -th column of Q k. D i 2D i (a) Suppos i, j, k ar all distinct. This cas follows from Lmma 3. (b) Suppos i = j k. W can choos a row in I k, 2I k and D i, 2D i, rspctivly. A row in I k, 2I k corrsponds to th columns 3
of f, f 2, f 3 has two sam lmnts and on diffrnt lmnt, anothr row corrsponds to th columns of f, f 2, f 3 hav two diffrnt lmnts in th corrsponding position of I k, 2I k with two sam lmnts. For xampl, th matrix f, f 2, f 3 xists a 3 3 subarray 2 2 α, whr α is any lmnt of GF (3). (c) Suppos i = j = k. W can choos two rows in D i, 2D i. Thn in th matrix f, f 2, f 3 thr xists 3 3 subarrays with th sam typs as in Cas (b). Cas 3 : Lt two columns f, f 2 b from Q i, f 3 is from Q j, i j. f, f 2 ar th i -th, j -th column of Q i, f 3 is th k -th column of Q j. Thn in th matrix f, f 2, f 3 xists a 3 3 subarrays with th sam typs of cas (b). Cas 4 : Choosing th matrix P i and two columns f, f 2 from Q j, th matrix P i, f, f 2 has a (k + 2) (k + 2) subarray with on of th following typs: T I k α T I k α 2 T I k α 2 whr α is any lmnt of GF (3), is a null vctor with ordr k. This matrix has rank k + 2. Cas 5 : Choosing th matrics P i, P j, a column f from Q k, th matrix T T P i, P j, f = I k I k b Di Dj Dkb whr b is a column of I k, 2I k, has rank 2k +. Hnc any thr factors of F ar linarly indpndnt. Acknowldgmnt Th work of A. Dy was supportd by th National Acadmy of Scincs, India undr th Snior Scintist program of th Acadmy. Th support is gratfully acknowldgd. 4
Rfrncs Chn, G.Z., Ji, L.J. and Li, J.G.(24). Th xistnc of mixd orthogonal arrays with four and fiv Factors of strngth two. J. Combin. Dsigns 22 323 342. Dy, A. and Mukrj, R. (999). Fractional Factorial Plans. Nw York: Wily. Hdayat, A. S., Sloan, N. J. A. and Stufkn, J. (999). Orthogonal Arrays: Thory and Applications. Nw York: Springr. Jiang, L. and Yin, J. X. (23). An approach of constructing mixd-lvl orthogonal arrays of strngth 3. Sci. China Math. 56 9 5 Nguyn, M. V. M. (28). Som nw constructions of strngth 3 mixd orthogonal arrays. J. Statist. Plann. Infrnc 38 22 233. Rao, C. R. (973). Som combinatorial problms of arrays and applications to dsign of xprimnts. In: A Survy of Combinatorial Thory (J.N.Srivastava, Ed.), Amstrdam: North-Holland, pp. 349 359. Sun, C. Y., Das, A. and Dy, A. (2). On th construction of asymmtric orthogonal arrays. Statist. Sinica 24 26. Sun, C. Y. and Dy, A. (23). Construction of asymmtric orthogonal arrays through finit gomtris. J. Statist. Plann. Infrnc 5 623 635. Sun, C. Y. and Kuhfld, W. F. (25). On th construction of mixd orthogonal arrays of strngth two. J. Statist. Plann. Infrnc 33 555 56. Zhang, T. F., Zong, Y. Y. and Dy, A. (26). On th construction of asymmtric orthogonal arrays. J. Statist. Plann. Infrnc 7 77 82. 5