ISSN: 2277-375 Constructon of Trend Free Run Orders for Orthogonal rrays Usng Codes bstract: Sometmes when the expermental runs are carred out n a tme order sequence, the response can depend on the run order. To avod unwanted tme effect, one may be nterested to select a run order n such a way that all effects are orthogonal to the trend effect. These types of desgns are nown as trend free desgns. There are many technques to construct trend free run orders. In ths paper, we construct some trend free run orders for symmetrc and asymmetrc orthogonal arrays usng the results due to [5], [8] and the property that any d-1 columns n the party chec matrx of a lnear [n,,d] q code are lnearly ndependent. Keyword: codes, Mnmum Dstance, Orthogonal arrays, Trend free run orders. I. INTRODUCTION In some expermental stuatons where the treatments are to be appled sequentally to expermental unts over space or tme, there may be an unnown or uncontrollable trend effect whch s hghly correlated wth the order n whch the observatons are obtaned. In such stuatons, one may prefer to assgn treatments to expermental unts n such a way that the usual estmates for the factoral effects of nterest are not affected by unnown trend. Such run orders are called trend free run orders. When trend free effects are consdered n factoral experments, the order of expermental runs s essental. Under such condtons t s helpful to use orthogonal arrays to arrange expermental runs and ther order smultaneously. For ths, one needs to derve the trend free property n the columns of array to gan an approprate order.[11]ntroduced the concept of orthogonal arrays n the context of fractonal factoral experments. Orthogonal arrays are related to combnatorcs, fnte felds, geometry and error-correctng codes. symmetrc orthogonal arrays, ntroduced by [10] have receved great attenton n recent years. Many researchers have dealt wth the problem of constructon of trend free desgns. See [], [12] and [5]. good deal of wor has been done on the constructon of trend free run orders of factoral desgns when all factors have same number of levels. [6], [2] and [9] consdered ths problem for mxed level factoral desgns. In ths paper, we construct some trend free run orders for symmetrc and asymmetrc orthogonal arrays, usng the property that any d-1 columns n the party chec matrx of a lnear [n,,d] q code are lnearly ndependent and the results due to [5] and [8] respectvely. Secton 1 gves the prelmnares requred. In secton 2 a bref ntroducton of codng theory s gven. Secton 3 presents the method to construct trend free run orders for symmetrc orthogonal arrays. Trend free run Poonam SnghPuja Thaplyal, Veena Budhraja orders for asymmetrc orthogonal arrays are constructed n Secton. II. PRELIMINRIES Defnton 1:n orthogonal array O(N,n,q 1 q 2 q n,g) of strength g, 2 g n s an N n matrx havng q ( 2) dstnct symbols n the th column, =1,2,,n such that n every N g sub matrx, all possble combnatons of symbols appear equally often as a row.in partcular, f q 1 =.. = q n = q, the orthogonal array s called symmetrc orthogonal arrayand s denoted by O(N,n,q,g), otherwse the array s called asymmetrc orthogonal array. Defnton 2:The system of orthogonal polynomals on m equally spaced ponts l=0,1,2,,m-1s the set of polynomals satsfyng where and s a polynomal of degree s. We assume that each polynomal n the system s scaled so that ts values are always ntegers. Defnton 3:(Trend vector t). The N values of a polynomal trend of degree s are the values of the orthogonal polynomal of degree s on N equally spaced ponts n defnton 2. From the above defnton, the lnear trend vector t can be expressed as follows: and 69
where ISSN: 2277-375 n the standard form, a correspondng party chec matrx s a normalzng constant. s gven as T H = [ In ] For a gven ordered allocaton of the treatments n d N (N run orthogonal desgn) to expermental unts,let denotes the ordered vector of observatons. Suppose these observatons are nfluenced by a tme trend that can be represented by a polynomal of degree s (1 s N-1). The model for d N can be wrtten as where s a N-vector of uncorrelated random errors wth zero means, s N n matrx of factor effect coeffcents (we consder only man effects for trend freeness) and T s the N matrx of polynomal trend coeffcents. The vectors and are the factor and trend parameter effects respectvely. run order s optmal for the estmaton of the factor effects of nterest polynomal n thepresenceof nusance s-degree ) trendf If(2.) s satsfed we say that the run order s s trend free. If x s any column of then the usual nner product nd t s any column of T s called the tme count between x and t. Crteron (2.) states that all the tme count are zero for optmal run order. III. CODING THEORY lnear [n,,d] q code C over GF(q),where q prme or prme power, n s the length, s the dmenson and d s the mnmum dstance, s a -dmensonal subspace of the n-dmensonal vector space V(n,q) over GF(q). The elements of C are called code words. The mnmum dstance d of the code s the smallest number of postons n whch two dfferent code words of C dffer. Equvalently, d s the smallest number of nonzero symbols n any nonzero codeword of C. lnear code may be concsely specfed by gvng a n generator matrx G whose rows form a bass for the code. The standard form of the generator matrx s G = [ I ] Where s an (n ) matrx wth entres from GF(q). The dual code C of an [n,,d] q code C s C = { v V(n,q)/ v.w=0 for all w C}. Ths s an [n,n-,d ] q code and an (n-) n generator matrx H of C s called a party chec matrx of C. If the generator matrx s gven 70 ny d -1 columns n generator matrx G of C are lnearly ndependent and any d-1 columns n party chec matrx H are lnearly ndependent. Trend free run orders for Symmetrc Orthogonal rrays In ths secton, we construct trend free run orders for symmetrc orthogonal arrays. [5]used Generalzed Fold over Scheme (GFS) to construct trend free desgns and also dscussed condtons for lnear trend free effects n GFS. These condtons nvolve the generator matrx. They also provded a method for constructon of generator matrx so that the systematc run order for a desgn s constructed by GFS. However ths method s dffcult to use. The generator matrces for the constructon of systematc run order are obtaned from lnear codes. We present here the man result due to [5] n the form of followng theorem. (For more detals see [7] Th.7.3.3). The method of constructon n Theorem 1 s called the Generalzed Foldover Technque. Theorem 1: Let q ( 2) be a prme or prme power. Suppose that there exsts an r n matrx M, wth elements from GF(q), such that every r g submatrx of M has ran g and every column of M has at least ( s+1) non zero elements. Then there exsts a symmetrc orthogonal array O( q r, n, q,g) n whch all man effects are s-trend free. We use the result due to [5] to construct some trend free run orders for orthogonal arrays obtanable from the lnear codes. Example 1: Consder the lnear code [8,,] 2 gven n [3] wth party chec matrx H where Consder those columns of H whch have weght 2, where, the weght of a column s defned as the number of nonzero elements n the column. We get the followng matrx M 1 1 1 1 1 1 M = 0 0 1 1 1 1 1 1 0 0 1 1 1 0 1 0 1 0 1 The matrx M satsfy the condton of Theorem 1 wth q=2, =, g= 3, s + 1 = 2, n = 7. Let ξ denote a vector wth entres from GF(2). Consderng all the 2 possble dstnct choces of ξ, we form a 2 7 array wth rows of the form.
= ISSN: 2277-375 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 0 1 1 1 0 1 0 1 0 1 The array generated s gven n Table I. From the table I, we observe that the tme counts for all the man effects are zero. Hence we get a trend free run orders for O(2,7,2,3) havng all the man effects lnear trend free. Example 2: Consder a lnear [10,5,] 2 code gven n [3]. The party chec matrx H of the lnear code s gven as From the matrx H f we retan all the columns wth weght 3, then we get the followng matrx L = The matrx L satsfy the condtons of Theorem 1 wth q=2, =5, g=3, s+1=3, n=5. Let ζ denote a 5 1 vector wth entres from GF(2). Consderng all the 2 5 possble dstnct choces of ζ we form 2 5 5 array wth rows of the form. trend free run order for O (2 5,5,2,3) s obtaned havng all man effects lnear and quadratc trend free. Table II lsts the trend free run orders for symmetrc orthogonal arrays (wth degrees of trend freeness of all man effects obtaned from a class of lnear codes gven by [3]. IV. TREND FREE RUN ORDERS FOR SYMMETRIC ORTHOGONL RRYS Trend free run orders for asymmetrc orthogonal arrays can also be obtaned from the party chec matrx of a lnear [n,,d] q code. Usng the generator matrces obtaned from lnear codes n Secton 3, we construct trend free run order for asymmetrc orthogonal arrays. Consder an O(N, n, q1 q 2 q n, g) whose columns are called as factors denoted by F 1, F 2,..., F n. lso consder GF(q), of order q, where q s a prme or prme power. For the factor F (1 n) defne u columns, say p, p,..., p, each of order 1 1 2 u wth elements from GF(q). Thus for the n factors we have n all columns. The followng result was proved n [1]. Theorem 2: Let M be the n matrx, where n = u such that any d 1 columns of C u and j j are lnearly ndependent. Then M can be parttoned as M= 1 2... n, where = [ p p... p u matrces of, ], 1 n. Then for each of the,..., g ; where g d 2, out 1, 2,..., n ; the matrx j g... has full column ran over GF(q), u1 u n Then an O( q, n, ( q ) ( q ), g) can be constructed. Usng Theorem 1 and Theorem 2 we present a method n the form of a theorem gven below. The proof of ths theorem follows from the proofs gven n [7] Th.7.3.3) and [1]. Theorem 3: Let q be a prme or prme power. If there exst an n matrx M wth entres from GF( q ) such that ) ny d-1 columns, where d n be any postve nteger, are lnearly ndependent over GF( q ) and ) Every column of M has at least s+1 non zero elements then u1 u n an O( q, n, ( q ) ( q ), g) can be constructed n whch all the man effects are trend free of order s.the method s explaned n the followng example. Example : Consder the matrx M obtaned n Example 1. Represent Mas [ M1 M2... M7 ], where M ; 1 7 th denotes the column of matrx M. To construct an orthogonal arrayo( 2, 6, ( 2 2 ) 2 5, 2) we choose the followng matrces, correspondng to the factors of the array. 1 1 = 0 0 = [ M 1 M 2 ], = M +1 2 5 1 0 The ran condton of the Theorem 2 s always satsfed for g = 2 by the above matrx M. Ths can also be shown wth above choces of matrces correspondng to the 6 () Let, j { 2, 3,..., 6} ; j. For ths choce of the ndces and j, the matrx, ] wll always have [ j ran 2, because any 3 columns of the matrx M are lnearly ndependent (Theorem 2). u 71
ISSN: 2277-375 () Let = 1 and j { 2, 3,, 6} ; j. For ths choce of ndces, j the matrx [, j ] wll always codes have ran 2 snce any 3 or fewer columns of M are lnearly ndependent. Thus n each case the ran condton of Theorem 2 s satsfed and the desred orthogonal array can be constructed by Computng where ξ s a 2 matrx whose rows are all possble -tuples over GF(2) and replacng the combnatons (00), (01), (10), (11) under the frst two columns by dstnct symbols 0, 1, 2, 3 respectvely we get an O( 2, 6, 2 ) ( 2 2 5 2).Here we observe that the run order for column (factor) wth symbols say F 1 s also lnear trend free wth the other remanng fve columns (factors). Thus we get trend free run order for asymmetrc orthogonal array O( 2,6, ( 2 2 ) 2 5, 2) n whch all the man effects are lnear trend free. In Table III we lst the trend free run orders (wth degree of trend free of all the man effects) for asymmetrc orthogonal arrays obtaned from a class of lnear codes gven by [3] Trend free symmetrc O s Degree of trend free for man effects n symmetrc O s [8,,] 2 (2,7,2,3) [10,5,] 2 (2 5,5,2,3) [12,6,] 2 (2 6,6,2,3) [16,8,5] 2 (2 8,8,2,) [18,9,6] 2 (2 9,9,2,5) [22,11,6] 2 (2 11,11,2,6) [2,12,7] 2 (2 12,12,2,6) [28,1,8] 2 (2 1,1,2,7) Table IO (2,7,2,3) B C D E F 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 1 0 1 0 0 0 0 1 1 0 1 0 1 0 0 1 0 1 0 1 1 0 1 1 0 1 0 1 0 0 1 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 0 0 0 0 1 0 1 1 [12,6,] 2 [16,8,5] 2 [18,9,6] 2 (2 5,3, (2 3 ) 2 2, 2) (2 6,5, (2 2 ) 2,2) (2 8,7, (2 2 ) 2 6,3) (2 8, 6, (2 3 ) 2 5, 2) (2 9,8, (2 2 ) 2 8,) (2 9, 7, (2 3 ) 2 6, 2) codes Tme count 0 0 0 0 0 0 symmetrc Orthogonal array Degree of trend free for man effects n symmetrc orthogonal array [22,11,6] 2 [2,12,7] 2 (2 11,10, (2 2 ) 2 9, ) (2 12,11, (2 2 ) 2 10,5) (2 12, 10, (2 3 ) 2 9, 3) [8,,] 2 (2,6, (2 2 ) 2 5,2) [10,5,5] 2 (2 5,, (2 2 ) 2 3,3) [28,1,8] 2 (2 1,13, (2 2 ) 2 12,6) (2 1, 12, (2 3 ) 2 11,) 72
ISSN: 2277-375 VI. CONCLUSION The generator matrces for the constructon of trend free run orders for orthogonal arrays can easly be obtaned from lnear codes as gven n ths artcle. Further ths technque can also be used to construct trend free run orders for orthogonal arrays of hgher/mxed levels and for hgher strength. REFERENCES [1] ggarwal, M.L. and Budhraja, V.(2003).Some New symmetrc Orthogonal rrays. Journal of Korean Statstcal Socety32: 3, 225-233. [2] Baley, R.., Cheng, C.S. and Kpns, P.(1992). Constructon of trend resstant factoral desgns. Statst. Snca. 2, 393-11. [3] Betsumya K. and Harada M. (2001b). Classfcaton of Formally Self-Dual Even Codes of length up to 16. Desgns, Codes and Cryptography. 23, 325-332. [] Cheng C.S. and D.M. Stenberg (1991). Trend robust two level factoral desgns. Bometra. 78, 325-336. [5] Coster D.C. and Cheng C.S. (1988). Mnmum cost trend free run orders of fractonal factoral desgn. The nnals of Statstcs. 16, 3, 1188-1205. [6] Coster, D.C. (1993). Trend free run orders of mxed level fractonal factoral desgns. nn. Statst..21, 2072-2086. [7] Dey,. and Muerjee, R. (1999). Fractonal Factoral Plans. Wley. New Yor. [8] Dey,., Das,. and Suen, Chung-y. (2001). On the Constructon of symmetrc Orthogonal rrays. StatstcaSnca11 (1), 21-260. [9] Jacroux, M. (199). On the constructon of trend resstant mxed level factoral run orders. nn. Statst 22, 90-916. [10] Rao, C.R. (1973). Some combnatoral problems of arrays and applcatons to desgn of experments. Survey of Combnatoral Theory (J.N.Srvastava ed.), 39-359, msterdam: North-Holland. [11] Rao, C.R. (197). Factoral Experments Dervable from Combnatoral rrangements of rrays. Journal of Royal Statstcal Socety (Suppl.), 9, 128-139. [12] Stenberg, D.M. (1988). Factoral experments wth tme trend. Technometrcs30, 259-269. 73