Necessary and sufficient conditions for some two variable orthogonal designs in order 44

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1 University of Wollongong Reserch Online Fculty of Informtics - Ppers (Archive) Fculty of Engineering n Informtion Sciences 1998 Necessry n sufficient conitions for some two vrile orthogonl esigns in orer 44 C. Koukouvinos Ntionl Technicl University of Athens, Greece M. Mitrouli University of Athens, Greece Jennifer Seerry University of Wollongong, jennie@uow.eu.u Puliction Detils This rticle ws originlly pulishe s Koukouvinos, C, Mitrouli, M n Seerry, J, Necessry n sufficient conitions for some two vrile orthogonl esigns in orer 44, Journl of Comintoril Mthemtics n Comintoril Computing, 28, 1998, Reserch Online is the open ccess institutionl repository for the University of Wollongong. For further informtion contct the UOW Lirry: reserch-pus@uow.eu.u

2 Necessry n sufficient conitions for some two vrile orthogonl esigns in orer 44 Astrct We give new lgorithm which llows us to construct new sets of sequences with entries from the commuting vriles 0, ±, ±, ± c, ± with zero utocorreltion function. We show tht for twelve cses if the esigns exist they cnnot e constrcte using four circulnt mtrices in the Goethls-Seiel rry. Further we show tht the necessry conitions for the existence of n OD(44; s1, s2) re sufficient except possily for the following 7 cses: (7, 32) (8, 31), (9, 30) (9, 33) (11,30) (13, 29) (15, 26) which coul not e foun ecuse of the lrge size of the serch spce for complete serch. These cses remin open. In ll we fin 398 cses, show 67 o not exist n estlish 12 cses cnnot e constructe using four circulnt mtrices. We give new construction for OD(2n) n OD(n + 1) from OD(n). The full OD(44; s1 s2, s3, 44 s1 s2 s3) given in this pper yiel t lest 68 equivlence clsses of Hmr mtrices. Keywors Autocorreltion, construction, sequence, orthogonl esign, AMS Suject Clssifiction: Primry 05B15, 05820, Seconry 62K05. Disciplines Physicl Sciences n Mthemtics Puliction Detils This rticle ws originlly pulishe s Koukouvinos, C, Mitrouli, M n Seerry, J, Necessry n sufficient conitions for some two vrile orthogonl esigns in orer 44, Journl of Comintoril Mthemtics n Comintoril Computing, 28, 1998, This journl rticle is ville t Reserch Online:

3 Necessry n sucient conitions for some two vrile orthogonl esigns in orer 44 C. Koukouvinos, M. Mitrouli y, n Jennifer Seerry z Deicte to Professor Anne Penfol Street Astrct We give new lgorithm which llows us to construct new sets of sequences with entries from the commuting vriles 0; ; ; c; with zero utocorreltion function. We show tht for twelve cses if the esigns exist they cnnot e constrcte using four circulnt mtrices in the Goethls-Seiel rry. Further we show tht the necessry conitions for the existence of n OD(44; s 1 ; s 2 ) re sucient except possily for the following 7 cses: (7; 32) (8; 31) (9; 30) (9; 33) (11; 30) (13; 29) (15; 26) which coul not e foun ecuse of the lrge size of the serch spce for complete serch. These cses remin open. In ll we n 398 cses, show 67 o not exist n estlish 12 cses cnnot e constructe using four circulnt mtrices. We give new construction for OD(2n) n OD(n + 1) from OD(n). The full OD(44; s 1 ; s 2 ; s 3 ; 44 s 1 s 2 s 3 ) given in this pper yiel t lest 68 equivlence clsses of Hmr mtrices. Key wors n phrses: Autocorreltion, construction, sequence, orthogonl esign. AMS Suject Clssiction: Primry 05B15, 05B20, Seconry 62K05. 1 Introuction Throughout this pper we will use the enition n nottion of Koukouvinos, Mitrouli, Seerry n Krels [2]. We note from [3] tht we hve to test 1 4 n2 = 484 cses. We n 398 cses, show 67 o not exist n estlish 12 cses cnnot e constructe using four circulnt mtrices. There re 7 open cses which coul not e foun ecuse of the lrge size of the serch spce for complete serch. 2 New orthogonl esigns Theorem 1 An OD(44; s1; s2) cnnot exist for the following 2 tuples (s1; s2): Deprtment of Mthemtics, Ntionl Technicl University of Athens, Zogrfou 15773, Athens, Greece. y Deprtment of Mthemtics, University of Athens, Pnepistemiopolis 15784, Athens, Greece. z School of IT n Computer Science, University of Wollongong, Wollongong, NSW, 2522, Austrli. 1

4 (1; 7) (1; 15) (1; 23) (1; 28) (1; 31) (1; 39) (1; 42) (2; 14) (2; 30) (3; 5) (3; 13) (3; 20) (3; 21) (3; 29) (3; 37) (3; 40) (4; 7) (4; 15) (4; 23) (4; 28) (4; 31) (4; 39) (5; 11) (5; 12) (5; 19) (5; 27) (5; 35) (6; 10) (6; 26) (7; 9) (7; 16) (7; 17) (7; 25) (7; 28) (7; 33) (7; 36) (8; 14) (8; 30) (9; 15) (9; 23) (9; 28) (9; 31) (10; 17) (10; 22) (10; 24) (11; 13) (11; 16) (11; 20) (11; 21) (11; 29) (12; 13) (12; 15) (12; 20) (12; 21) (12; 29) (13; 19) (13; 27) (14; 18) (15; 16) (15; 17) (15; 20) (15; 25) (16; 19) (16; 23) (16; 28) (17; 23) (19; 20) (19; 21) Proof. These cses re eliminte y the numer theoretic necessry conitions given in [1] or [2, Lemm 3]. Exmple. To illustrte how we use the numer theoretic conitions to estlish the nonexistence of n OD(4n; 11; 20) we consier the prouct = now this is numer of the form 4 (8 + 7) which cnnot e written s the sum of three squres n hence n OD(4n; 11; 20) cnnot exist. Remrk. A computer serch, which we elieve ws exhustive, ws crrie out which les us to elieve tht 1. there re no 4-NP AF (7; 19) sequences of length there re no 4-NP AF (3; 31), 4-NP AF (5; 30), 4-NP AF (6; 29) n 4-NP AF (8; 27) sequences of length 9. This mens tht there re lso no 4-N P AF (1; 5; 30), 4- NP AF (1; 6; 29) n 4-NP AF (1; 8; 27) of length there re no 4-NP AF (2; 41) sequences of length 11. This mens tht there re lso no 4-NP AF (1; 2; 41) sequences of length there re no 4-NP AF (6; 37) sequences of length 11. Lemm 1 OD(44; 1; 1; 42) n n OD(44; 1; 3; 40) o not exist (this is prove theoreticlly). The Germit-Verner Theorem sys tht if n OD(44; 3; 40) exists then n OD(44; 1; 3; 40) will exist, n if n OD(44; 1; 42) exists then n OD(44; 1; 1; 42) will exist. Hence the OD(44; 1; 42) n OD(44; 3; 40) o not exist. Lemm 2 The following OD(44; 1; ; 43 ) n OD(44; ; 43 ) cnnot e constructe using four circulnt mtrices in the Goethls-Seiel rry: (6; 37) (1; 6; 37) (10; 33) (1; 10; 33) (12; 31) (1; 12; 31) (13; 30) (1; 13; 30) (14; 29) (1; 14; 29) (16; 27) (1; 16; 27) (19; 24) (1; 19; 24) (20; 23) (1; 20; 23) Proof. By the Germit-Verner theorem if n orthogonl esign OD(n; x1; x2; ; x u 1; x u ) with u i=1 x i = n 1 exists, n 0(mo 4), then n OD(n; 1; x1; x2; ; x u 1; x u ) exists. Now for ech of the cses in this lemm we hve n OD(44; ; 43 ) n tht is y the Germit-Verner theorem n OD(44; 1; ; 43 ). Using the sum-ll mtrix metho we write 1 = , = n 43 = We require the sum-ll mtrix to e 34 orthogonl mtrix with the rst row contining 1; 0; 0; n 0; the secon row contining 1; 2; 3; n 4; in some orer n the thir row contining 1; 2; 3; n 4; in some orer. 2

5 However, s we illustrte for OD(1; 20; 23), this is not possile for the cses mentione in the enuncition. Using the sum-ll mtrix metho for OD(1; 20; 23), 1 = , 20 = n 23 = ( 1) There is no wy to form n orthogonl mtrix unless oth 20 n 23 cn e written s the sum of 3 squres. 2 Theorem 2 There re OD(44; s1; s2; s3; 44 s1 s2 s3) constructe using four sequences to otin four circulnt mtrices for use in the Goethls-Seiel rry for the following 2 tuples: 1; 43 1; 2; 41 2; 2; 4; 36 2; 2; 8; 32 2; 2; 20; 20 2; 6; 12; 16 2; 8; 16; 16 5; 39 7; 37 1; 9; 34 1; 11; 32 13; 31 15; 29 1; 17; 26 1; 18; 25 21; 23 Corollry 1 By suitly choosing the vriles of the known OD(44; s1; s2; s3; 44 s1 s2 s3) to e replce y 1 these le to t lest 36 lgericlly inequivlent Hmr mtrices of orer 44. By suitly choosing the vriles of the known OD(44; s1; 44 s1) to e replce y 1 these le to t lest 12 more lgericlly inequivlent Hmr mtrices of orer 44. Corollry 2 By suitly choosing the vriles of the known OD(44; 1; s1; 35 s1) to e replce y 1 we otin t lest 20 lgericlly inequivlent skew-hmr mtrices of orer 44. The numer epens on whether ech skew-hmr mtrix is equivlent to its trnspose or not. 3 New Algorithm The lgorithm previously use to n OD vi four sequences of length t 10 ws prohiitively slow for length 11. Hence we trie new lgorithm, which epene on the previous lgorithm, to n rst W (4t; k) me with four sequences of length t with P AF = 0 or NP AF = 0. In the new lgorithm MAT LAB ws use to set up series of equtions to e solve for ech iniviul k n then ll solutions to these equtions were foun. Exmple. We illustrte the lgorithm y trying to construct the OD(44; 11; 27). We rst notice tht 11 hs unique ecomposition into squres 11 = , while 27 hs three ecompositions into four squres. All three cn e use in this construction s they must e le to e use in n integer mtrix (the sum-ll mtrix) which is orthogonl. Hence we use 27 = = = So we hve the mtrices " # ; " # or " We now ll ech of the positions which re represente y 0 y one of 17 vriles x1; x2; ; x17. We now use MAT LAB to expn the rst rows to mke four circulnt mtrices with row inner prouct zero: this correspons to forming four sequences with P AF = 0. The equtions will e those tht involve some x j, 1 j 17 with, n those which hve no terms in. This gives t most 6 inepenent equtions. A serch is now me through the 17 vriles, llowing them to ssume the vlues 0; 1, where six of them must lwys e zero, n using the extr constrints tht 3 #

6 3X 5X x i = 1; x i X11 = 1; x i X17 = 3; x i = 0: i=1 i=4 i=6 i=12 We strt with the following four sequences of length 11 n PAF = We replce the 1 y vrile such s n we replce the 17 zeros y the vriles. Thus we hve the sequences x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 We then use MATLAB to set up series of equtions, tht when solve, yiel, mong others, the following solution: x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x We now replce the vriles in the originl four sequences y these solutions to otin the OD(44; 11; 27). 2 Remrk. Using this lgorithm we teste ll unknown two vrile cses n foun 7 cses which we were unle to resolve ue to the extremely lrge serch spce. We estimte tht complete serch for the OD(44; 7; 32) using this lgorithm requires 2 37 opertions. 2 4 New Results Theorem 3 Write X(; ) = fe1x1; e2x2; ; e n 1x n 1; e n x n g, Y (; ) = ff1y1; f2y2; ; f n 1y n 1; f n y n g for the sequences of length n, NPAF = 0, where e i n f i re chosen from ; where, re commuting vriles n x i, y i hve elements 0; 1 n the sequences X(1; 1) n Y (1; 1) hve NPAF = 0. Suppose occurs totl of s1 times n totl of s2 times then we sy the two sequences we hve re 2-NP AF (n; s1; s2). Write i = if e i = n i = if e i = for i = 1,...,n, n similrly, i = if f i = n i = if f i = for i = 1,...,n. Then (i) X(; ); Y (; ) n Y (; ); X ( ; ) where Z enotes the reverse of the sequence Z or n fe1x1; e2x2; ; e n 1x n 1; e n x n ; n y n ; n 1y n 1; ; 2y2; 1y1g ff1y1; f2y2; ; f n 1y n 1; f n y n ; n x n ; n 1x n 1; ; 2x2; 1x1g re two sequences with elements f0; ; g with NPAF = 0. These sequences re 2-NP AF (2n; 2s1; 2s2). 4

7 n (ii) If x n 1 n y n 1 re oth zero then the sequences fe1x1; e2x2; ; n y n ; e n x n ; n 2y n 2; ; 2y2; 1y1g ff1y1; f2y2; ; n x n ; f n y n ; n 2x n 2; ; 2x2; 1x1g re two sequences with elements f0; ; g with NPAF = 0. These sequences re 2-NP AF (2n 2; 2s1; 2s2). (iii) Similrly with 4-NP AF (n; s1; s2), X(; ), Y (; ), Z(; ) n W (; ) we hve X(; ); Y (; ); Y (; ); X ( ; ); Z(; ); W (; ) n W (; ); Z ( ; ) where Z enotes the reverse of the sequence Z re 4-NP AF (2n; 2s1; 2s2). (iv) Similrly with 4-NP AF (n; s1; s2), if the secon lst element of ech of the four sequences is zero then proceeing s in (ii) we otin 4-NP AF (2n 2; 2s1; 2s2). (v) Similrly if there re 4-NP AF (n; s1; s2), n the secon lst element of two of the sequences is zero n the lst element of two of the sequences is zero then comining the methos of (ii) n (iii) we cn get 4-NP AF (2n 2; 2s1; 2s2). Proof. The proof follows y writing out the sequences n checking the NPAF. Exmple. We use to men n c to men c. To illustrte prt (v) of the theorem we note tht c c c 0 c 0 c c c 0 c 0 c 0 c 0 c 0 c c 0 c 0 c 0 c n c c c c 0 c c c c 0 c 0 c 0 c 0 c c 0 c 0 c 0 c re 4-NP AF (7; 2; 16) n 4-NP AF (7; 4; 16), respectively. In fct we note c c c 0 c 0 c c c 0 c 0 c 0 c c 0 c c 0 c 0 c 0 c n c c c c 0 c c c c 0 c 0 c c 0 c c 0 c 0 c 0 c re 4-NP AF (7; 1; 2; 16) n 4-NP AF (7; 1; 4; 16), respectively. We lso note tht c c c c c c 0 c 0 c c c c c c c 0 c 0 c c c c c c c c 0 c c c c c c c c 0 c n c c c c c c c c c c c c c c c c c 0 c 0 c c c c c 0 c c 0 c 0 c c c c c 0 c re 4-NP AF (11; 2; 4; 32) n 4-NP AF (12; 2; 8; 32), respectively. c c c c c c 0 c 0 c c c c c c c 0 c 0 c c c c c c c 0 c 0 c c c c c c c 0 c 0 c n c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c 5

8 re 4-NP AF (11; 4; 32) n 4-NP AF (12; 8; 32), respectively. Lemm 3 If there exist 2-NP AF (n; s1; s2) then there exist 4-NP AF (n + 1; 2; 2; 2s1; 2s2). Corollry 3 Since there exist 2-NP AF (n; s1; s2) for the vlues liste in the tle we get the corresponing lrger 4-NP AF (n + 1; 2; 2; 2s1; 2s2). 2-NP AF (n; s1; s2) ) 4-NP AF (n + 1; 2; 2; 2s1; 2s2) (9;13) (10;2,2,26) (11;13) (12;2,2,26) (14;17) (15;2,2,34) (18;25) (19;2,2,50) (4;4,4) (5;2,2,8,8) (6;2,8) (7;2,2,4,16) (6;5,5) (7;2,2,10,10) (8;8,8) (9;2,2,16,16) (10;10,10) (11;2,2,20,20) (14;13,13) (15;2,2,26,26) Corollry 4 Using the previous theorem we see tht 4-NP AF (n; s1; s2) ) 4-NP AF (2n; 2s1; 2s2) NPAF(5;1,18) NPAF(5;1,19) NPAF(5;2,17) NPAF(5;2,18) NPAF(5;3,17) NPAF(7;3,18) NPAF(5;4,16) NPAF(7;4,17) NPAF(7;4,18) NPAF(5;5,14) NPAF(5;5,15) NPAF(7;5,16) NPAF(7;5,17) NPAF(7;5,18) NPAF(5;6,14) NPAF(7;6,16) NPAF(7;7,14) NPAF(7;7,15) NPAF(5;8,11) NPAF(5;8,12) NPAF(5;9,10) NPAF(5;9,11) NPAF(7;9,12) NPAF(10;2,36) NPAF(10;2,38) NPAF(10;4,34) NPAF(10;4,36) NPAF(10;6,34) NPAF(14;6,36) NPAF(10;8,32) NPAF(14;8,34) NPAF(14;8,36) NPAF(10;10,28) NPAF(10;10,30) NPAF(14;10,32) NPAF(14;10,34) NPAF(14;10,36) NPAF(10;12,28) NPAF(14;12,32) NPAF(14;14,28) NPAF(14;14,30) NPAF(10;16,22) NPAF(10;16,24) NPAF(10;18,20) NPAF(10;18,22) NPAF(14;18,24) Theorem 4 The sequences given in the Appenices cn e use to construct the pproprite esigns to estlish tht the necessry conitions for the existence of n OD(44; s1; s2) re sucient, except possily for the following 12 cses which cnnot e constructe from four circulnt mtrices: 6

9 (5; 38) (6; 37) (8; 35) (10; 33) (12; 31) (13; 30) (14; 29) (15; 28) (16; 27) (19; 24) (20; 23) (21; 22): n the following 7 cses which re unecie: (7; 32) (8; 31) (9; 30) (9; 33) (11; 30) (13; 29) (15; 26) Remrk. There re 484 possile 2 tuples. Tle 1 lists the 398 which correspon to esigns which exist in orer 44: 67 2-tuples correspon to esigns eliminte y numer theory (NE). For 12 cses, if the esigns exist, they cnnot e constructe using circulnt mtrices (Y). 7 cses remin unecie. P inictes tht 4-P AF sequences with length 11 exist; n inictes 4-N P AF sequences with length n exist. 7

10 N E N E N E N E N E N E N E N E N E ; ; P N E N E N E 3 21 N E N E N E N E N E N E N E N E N E ; P N E N E 5 12 N E N E N E P 5 35 N E P 5 38 Y N E N E P P; P Y N E N E 7 17 N E N E N E P N E 7 34 P 7 35 P 7 36 N E N E P P 8 30 N E P Y N E N E Tle 1: The existence of OD(44; s 1; s 2). 8

11 N E 9 29 P N E 9 32 P; P 9 35 P N E P N E N E P P P Y N E N E N E N E P P P N E P P P N E N E N E N E N E P P N E P; Y N E P P P P N E P Y P P P N E P P P P P; Y P N E N E N E P P P N E P; Y P N E N E P Y N E P P N E P P P P P P P P P P N E N E P P Y P P Y Y P Tle 1(Cont): The existence of OD(44; s 1; s 2). References [1] A.V.Germit, n J.Seerry, Orthogonl esigns: Qurtic forms n Hmr mtrices, Mrcel Dekker, New York-Bsel, [2] C.Koukouvinos, M.Mitrouli, J.Seerry, n P.Krels, On sucient conitions for some orthogonl esigns n sequences with zero utocorreltion function, Austrls. J. Comin., 13, (1996), [3] C.Koukouvinos n Jennifer Seerry, New orthogonl esigns n sequences with two n three vriles in orer 28, Ars Comintori, (to pper). 9

12 Design 4; 10; 10) (1; A 1; A 2 c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c 0 c c c c c 0 c c c 0 0 A 3; A 4 c c c c c c c 0 c c c c c c c c 0 c c c c c c 0 c 0 c c c c c c c c c c c c c 0 c c c c c c 0 c c c c Appenix A: Orer 40 (Sequences with zero non-perioic utocorreltion function) notinyet (1; 4; 32) (2; 2; 18; 18) (2; 2; 34) (2; 4; 32) (2; 4; 16; 18) (2; 10; 10; 18) 10 (2; 12; 22) (2; 35) (3; 31) (3; 34)

13 Design 8; 8; 16) (4; A 1; A 2 c c c c c c c c c c c c c c c c c c c c c 0 c c c c 0 c c c c c c c A 3; A 4 c c c c c c c c c c c c c c c c 0 c 0 c 0 c 0 c c c c c c c c 0 c c c c c c c Appenix A(cont): Orer 40 (Sequences with zero non-perioic utocorreltion function) notinyet (4; 4; 16; 16) (4; 6; 12; 18) (4; 8; 8; 16) (4; 10; 10; 16) (5; 30) (5; 33) 11 (6; 31) (7; 31) (8; 8; 10; 10) (10; 10; 10; 10)

14 Design 4; 16; 16) (1; A 1; A 2 0 c c 0 c c 0 c 0 c 0 c c c c c c c c 0 0 c c c 0 0 c c c A 3; A 4 c 0 c c c c 0 c c c c c c c c c 0 0 c c c 0 0 c c c Appenix B: Orer 44 (Sequences with zero non-perioic utocorreltion function) (1; 30) (1; 34) (1; 35) (1; 37) (1; 38) (1; 40) 12 (1; 41) (2; 2; 4; 36) (2; 2; 8; 32) (2; 2; 20; 20) (2; 6; 12; 16)

15 Design 8; 16; 16) (2; A 1; A 2 c c c 0 c c c c 0 c A 3; A 4 c c c c c c c c Appenix B(cont): Orer 44 (Sequences with zero non-perioic utocorreltion function) (2; 37) (2; 39) (3; 35) (3; 36) (3; 38) (3; 39) (3; 41) (4; 35) 13 (5; 36) (5; 39) (7; 37)

16 Design 9; 34) (1; A 1 A 2 c c c c A 3 A 4 c 0 c Appenix C: Orer 44 (Sequences with zero perioic utocorreltion function) (1; 11; 32) (1; 17; 26) (1; 18; 25) (2; 12; 27) (2; 41) (4; 37) (5; 34) (5; 37) 14 (6; 29) (6; 33) (6; 35) (7; 30)

17 Design 34) (7; A 1 A A 3 A Appenix C(cont): Orer 44 (Sequences with zero perioic utocorreltion function) (7; 35) (8; 27) (8; 29) (8; 33) (9; 32) (10; 31) (11; 27) (11; 28) 15 (11; 31) (12; 25) (12; 26) (12; 30)

18 Design 22) (13; A 1 A A 4 A Appenix C(cont): Orer 44 (Sequences with zero perioic utocorreltion function) 0 0 (13; 24) (13; 25) (13; 26) (13; 28) (13; 31) (14; 15) (14; 17) (14; 23) 16 (14; 24) (14; 25) (14; 28)

19 Design 30) (14; A 1 A A 3 A Appenix C(cont): Orer 44 (Sequences with zero perioic utocorreltion function) (15; 22) (15; 23) (15; 24) (15; 27) (15; 29) (17; 21) (17; 22) (17; 24) 17 (17; 25) (18; 19) (18; 21)

20 Design 23) (18; A 1 A A 3 A Appenix C(cont): Orer 44 (Sequences with zero perioic utocorreltion function) (19; 22) (19; 23) (20; 21) (21; 23) 18