On Full Orthogonal Designs in Order 56

We find new full orthogonal designs in order 56 and show that of 1285 possible OD(56; s1, s2, s3, 56—s1—s2-s3) 163 are known, of 261 possible OD(56; s1, s2, 56—s1—s2) 179 are known. All possible OD(56; s1, 56 — s1) are known. Keywords : Construction, sequences, circulant matrices, amicable sets, orthogonal designs, AMS Subject Classification: Primary 05B15, 05B20, Secondary 62K05. Publication Details This article was originally published as Georgiou, S, Koukouvinos, C and Seberry, J, On Full Orthogonal Designs in Order 56, Ars Combinatoria 65, 2002, 79-89. This journal article is available at Research Online: http://ro.uow.edu.au/infopapers/314 On Full Orthogonal Designs in Order 56 S. Georgiou , C. Koukouvinos , and Jennifer Seberryy September 1, 2000 Abstra t We nd new full orthogonal designs in order 56 and show that of 1285 possible OD(56; s1; s2; s3; 56 s1 s2 s3) 163 are known, of 261 possible OD(56; s1; s2; 56 s1 s2) 179 are known. All possible OD(56; s1; 56 s1) are known. Key words and phrases: Constru tion, sequen es, ir ulant matri es, ami able sets, orthogonal designs. AMS Subje t Classi ation: Primary 05B15, 05B20, Se ondary 62K05. 1 Introdu tion An orthogonal design of order n and type (s1; s2; : : : ; su) (si > 0), denotedOD(n; s1; s2; : : : ; su), on the ommuting variables x1; x2; : : : ; xu is an n nmatrixA with entries from f0; x1; x2; : : : ; xug su h that AAT = ( u Xi=1 six2i )In Alternatively, the rows of A are formally orthogonal and ea h row has pre isely si entries of the type xi. In [2℄, where this was rst de ned, it was mentioned that ATA = ( u Xi=1 six2i )In and so our alternative des ription of A applies equally well to the olumns of A. It was also shown in [2℄ that u (n), where (n) (Radon's fun tion) is de ned by (n) = 8 +2d, when n = 2ab, b odd, a = 4 + d, 0 d < 4. A weighing matrix W = W (n; k) is a square matrix with entries 0; 1 having k nonzero entries per row and olumn and inner produ t of distin t rows zero. Hen e W satis es WW T = kIn, and W is equivalent to an orthogonal design OD(n; k). The number k is alled the weight of W . If k = n, that is, all the entries of W are 1 and WW T = nIn, then W is alled an Hadamard matrix of order n.In this ase n = 1; 2 or n 0(mod 4). Department of Mathemati s, National Te hni al University of Athens, Zografou 15773, Athens, Gree e. yDepartment of Computer S ien e, University of Wollongong, Wollongong, NSW, 2522, Australia. 1 Given the sequen e A = fa1; a2; : : : ; ang of length n the non-period i auto orrelation fun tion NA(s) is de ned as NA(s) = n s Xi=1 aiai+s; s = 0; 1; : : : ; n 1; (1) If A(z) = a1 + a2z + : : :+ anzn 1 is the asso iated polynomial of the sequen e A, then A(z)A(z 1) = n Xi=1 n X j=1 aiajzi j = NA(0) + n 1 X s=1NA(s)(zs + z s); z 6= 0: (2) Given A as above of length n the periodi auto orrelation fun tion PA(s) is de ned, redu ing i+ s modulo n, as PA(s) = n Xi=1 aiai+s; s = 0; 1; : : : ; n 1: (3) The following theorem whi h uses four ir ulant matri es in the Goethals-Seidel array is very useful in our onstru tion for orthogonal designs. Theorem 1 [3, Theorem 4.49℄ Suppose there exist four ir ulant matri es A, B, C, D of order n sati sfying AAT +BBT +CCT +DDT = fIn Let R be the ba k diagonal matrix. Then GS = 0BBB A BR CR DR BR A DTR CTR CR DTR A BTR DR CTR BTR A 1CCCA is aW (4n; f) when A, B, C, D are (0; 1; 1) matri es, and an orthogonal design OD(4n; s1; s2; : : : ; su) on x1; x2; : : : ; xu when A, B, C, D have entries from f0; x1; : : : ; xug and f = Puj=1(sjx2j). 2 Corollary 1 If there are four sequen es A, B, C, D of length n with entries from f0; x1; x2; x3; x4g with zero periodi or non-periodi auto orrelation fun tion, then these sequen es an be used as the rst rows of ir ulant matri es whi h an be used in the GoethalsSeidel array to form an OD(4n; s1; s2; s3; s4). We note that if thew non-periodi auto orrelation fun tion is zero, then there are sequen es of length n+m for all m 0. 2 This method for onstru ting orthogonal designs was used in [1, 5℄. Throughought this paper we will use the de nition and notation of Koukouvinos, Mitrouli, Seberry and Karabelas [5℄. A pair of matri es A;B is said to be ami able (anti-ami able) if ABT BAT = 0 (ABT + BAT = 0). Following [7℄ a set fA1; A2; : : : ; A2ng of square real matri es is said to be ami able if n Xi=1 A (2i 1)AT (2i) A (2i)AT (2i 1) = 0 (4) 2 for some permutation of the set f1; 2; : : : ; 2ng. For simpli ity, we will always take (i) = i unless otherwise spe i ed. Son Xi=1 A2i 1AT2i A2iAT2i 1 = 0: (5) Clearly a set of mutually ami able matri es is ami able, but the onverse is not true in general. Throughout this paper Rk denotes the ba k diagonal identity matrix of order k. Let fAig8i=1 be an ami able set of ir ulant matri es of order t, satisfying the additive property for (s1; s2; : : : ; sk). Then the Kharaghani array H = BBBBBBBBBBBB A1 A2 A4Rn A3Rn A6Rn A5Rn A8Rn A7Rn A2 A1 A3Rn A4Rn A5Rn A6Rn A7Rn A8Rn A4Rn A3Rn A1 A2 AT8Rn AT7Rn AT6Rn AT5Rn A3Rn A4Rn A2 A1 AT7Rn AT8Rn AT5Rn AT6Rn A6Rn A5Rn AT8Rn AT7Rn A1 A2 AT4Rn AT3Rn A5Rn A6Rn AT7Rn AT8Rn A2 A1 AT3Rn AT4Rn A8Rn A7Rn AT6Rn AT5Rn AT4Rn AT3Rn A1 A2 A7Rn A8Rn AT5Rn AT6Rn AT3Rn AT4Rn A2 A1 CCCCCCCCCCCCA is an OD(8t; s1; s2; : : : ; sk). The Kharaghani array whi h uses ami able sets of eight matri es is also very useful in our onstru tions for orthogonal designs. The following lemma applies a lemma given in Georgiou, Koukouvinos, Mitrouli and Seberry [1℄ to determine the number of possible tuples to be sear hed determining the size of sear h spa e for orthogonal designs in order 56. Lemma 1 Let n = 4m = 56 be the order of an orthogonal design then the number of ases whi h must be studied to determine whether all orthogonal designs exist is (i) 1 4n2 = 784 when 2 tuples are onsidered; (ii) n 2 72 (2n2 + 7n+ 6) = 5004 when 3-tuples are onsidered; (iii) 1 576 (n4 + 6n3 2n2 24n+ 64) = 18890 when 4 tuples are onsidered. 2 New full orthogonal designs from smaller orders Theorem 2 There are OD(56; s1; s1; 65 s1; 56 s1) onstru ted using the full OD(28; s1; 28 s1) given in [2; 5; 6℄ for: (1; 1; 27; 27) (2; 2; 26; 26) (3; 3; 25; 25) (4; 4; 24; 24) (5; 5; 23; 23) (6; 6; 22; 22) (7; 7; 21; 21) (8; 8; 20; 20) (9; 9; 19; 19) (10; 10; 18; 18) (11; 11; 17; 17) (12; 12; 16; 16) (13; 13; 15; 15) (14; 14; 14; 14) Proof. We use the ami able orthogonal designs of type AOD(2; (1; 1); (1; 1)) in order two with the two variable designs in order 28 to obtain the desired designs in order 56. 2 3 Theorem 3 There are full OD(56; s1; s2; s3; 56 s1 s2 s3) onstru ted using the full OD(28; s1; s2; 28 s1 s2) and OD(28; s1; s2; s3; 28 s1 s2 s3) designs in order 28 for the 4-tuples given in Table 2 (1; 1; 2; 52) (1; 1; 4; 50) (1; 1; 6; 48) (1; 1; 12; 42) (1; 1; 16; 38) (1; 1; 18; 36) (1; 1; 26; 28) (1; 2; 2; 51) (1; 2; 3; 50) (1; 2; 16; 37) (1; 2; 17; 36) (1; 2; 26; 27) (1; 3; 16; 36) (1; 3; 26; 26) (1; 6; 12; 37) (1; 6; 13; 36) (1; 7; 12; 36) (1; 18; 18; 19) (2; 2; 2; 50) (2; 2; 8; 44) (2; 2; 13; 39) (2; 2; 14; 38) (2; 2; 16; 36) (2; 2; 18; 34) (2; 2; 25; 27) (2; 2; 26; 26) (2; 3; 3; 48) (2; 3; 12; 39) (2; 3; 15; 36) (2; 4; 25; 25) (2; 6; 6; 42) (2; 6; 12; 36) (2; 6; 18; 30) (2; 6; 24; 24) (2; 8; 8; 38) (2; 8; 10; 36) (2; 9; 9; 36) (2; 9; 18; 27) (2; 12; 18; 24) (2; 12; 21; 21) (2; 13; 13; 28) (2; 13; 15; 26) (2; 14; 14; 26) (2; 16; 18; 20) (2; 16; 19; 19) (2; 18; 18; 18) (3; 3; 12; 38) (3; 3; 14; 36) (3; 3; 20; 30) (3; 5; 12; 36) (4; 4; 4; 44) (4; 4; 8; 40) (4; 4; 12; 36) (4; 4; 16; 32) (4; 4; 20; 28) (4; 7; 7; 38) (4; 8; 8; 36) (4; 8; 12; 32) (4; 8; 18; 26) (4; 8; 20; 24) (4; 8; 22; 22) (4; 9; 9; 34) (4; 12; 20; 20) (4; 13; 13; 26) (4; 14; 19; 19) (4; 16; 18; 18) (4; 17; 17; 18) (5; 5; 10; 36) (5; 5; 18; 28) (5; 10; 18; 23) (5; 15; 18; 18) (6; 6; 6; 38) (6; 6; 8; 36) (6; 7; 7; 36) (6; 10; 10; 30) (6; 12; 18; 20) (6; 12; 19; 19) (6; 14; 18; 18) (6; 15; 15; 20) (7; 7; 14; 28) (7; 14; 14; 21) (8; 8; 8; 32) (8; 8; 10; 30) (8; 8; 16; 24) (8; 8; 18; 22) (8; 8; 20; 20) (8; 10; 10; 28) (8; 10; 18; 20) (8; 12; 18; 18) (8; 14; 14; 20) (8; 16; 16; 16) (9; 9; 10; 28) (9; 9; 18; 20) (9; 10; 10; 27) (9; 10; 18; 19) (9; 11; 18; 18) (10; 10; 16; 20) (10; 10; 18; 18) (10; 14; 14; 18) (14; 14; 14; 14) Table 1: Full 4-variable OD(56; s1; s2; s3; 56 s1 s2 s3) onstru ted from full three and four variable designs in order 28. Theorem 4 There are OD(56; s1; s1; 2s2; 2s3; 56 2s1 2s2 2s3) onstru ted using the Multipli ation Theorem [3, Lemma 4.11℄ with the full OD(28; s1; s2; s3; 28 s1 s2 s3) given in [2; 5; 6℄ for the values given in Table 2: (1; 1; 2; 2; 50) (1; 1; 2; 16; 36) (1; 1; 2; 26; 26) (1; 1; 6; 12; 36) (1; 1; 18; 18; 18) (2; 2; 2; 25; 25) (2; 2; 8; 8; 36) (2; 2; 13; 13; 26) (2; 2; 16; 18; 18) (2; 3; 3; 12; 36) (2; 6; 6; 6; 36) (2; 6; 12; 18; 18) (2; 9; 9; 18; 18) (4; 4; 4; 8; 36) (4; 4; 8; 8; 32) (4; 4; 8; 20; 20) (4; 8; 8; 18; 18) (5; 5; 10; 18; 18) (7; 7; 14; 14; 14) (8; 8; 8; 16; 16) (8; 8; 10; 10; 20) (9; 9; 10; 10; 18) Table 2: Full 5-variable designs in order 56 from full 4-variable designs in order 28. In table 3 we present the new ami able sets of eight matri es whi h an be used in the Kharaghani array to onstru t some new full orthogonal designs in order 56. 4 A1 A2 Type A3 A4 ZERO A5 A6 A7 A8 (1,1,25,29) ( a; d; d; d; d; d; d ) ( b; b; b; b; b; b; b ) PAF ( b; d; d; d; d; d; d ) ( b; d; d; d; d; d; d ) n=7 ( d; d; d; d; d; d; d ) ( ; b; b; b; b; b; b ) ( b; b; b; b; b; b; b ) ( b; b; b; b; b; b; b ) (1,2,3,25,25) ( a; d; d; d; d; d; d ) ( a; d; d; d; d; d; d ) PAF ( a; d; d; d; d; d; d ) ( h; h; h; h; h; h; h ) n=7 ( d; d; d; d; d; d; d ) ( g; h; h; h; h; h; h ) ( e; h; h; h; h; h; h ) ( e; h; h; h; h; h; h ) (1,2,8,45) ( a; b; b; a; b; a; a ) ( b; b; b; b; b; b; b ) PAF ( a; b; b; a; b; a; a ) ( b; b; b; b; b; b; b ) n=7 ( d; b; b; b; b; b; b ) ( d; b; b; b; b; b; b ) ( ; b; b; b; b; b; b ) ( b; b; b; b; b; b; b ) (1,2,13,40) ( a; b; b; a; b; a; a ) ( a; a; a; a; a; a; a ) PAF ( b; a; a; b; a; b; b ) ( a; a; a; a; a; a; a ) n=7 ( ; a; a; a; a; a; a ) ( ; a; a; a; a; a; a ) ( d; b; b; b; b; b; b ) ( a; a; a; a; a; a; a ) (1,2,14,39) ( a; b; b; b; b; b; b ) ( d; a; a; a; a; a; a ) PAF ( d; b; b; b; b; b; b ) ( b; b; b; b; b; b; b ) n=7 ( ; b; b; b; b; b; b ) ( b; b; b; b; b; b; b ) ( b; a; a; b; a; b; b ) ( a; b; b; a; b; a; a ) (1,2,19,34) ( a; b; b; a; b; a; a ) ( a; b; b; a; b; a; a ) PAF ( ; b; b; b; b; b; b ) ( a; a; a; a; a; a; a ) n=7 ( d; b; b; b; b; b; b ) ( a; a; a; a; a; a; a ) ( b; a; a; a; a; a; a ) ( ; a; a; a; a; a; a ) (1,3,8,19,25) ( a; b; b; a; b; a; a ) ( e; h; h; h; h; h; h ) PAF ( a; b; b; a; b; a; a ) ( e; h; h; h; h; h; h ) n=7 ( e; h; h; h; h; h; h ) ( b; b; b; b; b; b; b ) ( d; b; b; b; b; b; b ) ( h; h; h; h; h; h; h ) (1,3,13,14,25) ( a; d; d; d; d; d; d ) ( e; f; f; e; f; e; e ) PAF ( f; e; e; f; e; f; f ) ( a; d; d; d; d; d; d ) n=7 ( f; f; f; f; f; f; f ) ( a; d; d; d; d; d; d ) ( d; d; d; d; d; d; d ) ( g; e; e; e; e; e; e ) (1,10,18,27) ( a; d; d; d; d; d; d ) ( d; b; b; b; b; b; b ) PAF ( a; d; d; d; d; d; d ) ( d; b; b; b; b; b; b ) n=7 ( ; d; d; d; d; d; d ) ( d; d; d; d; d; d; d ) ( a; b:b; a; b; a; a ) ( a; b; b; a; b; a; a ) Table 3: New full orthogonal designs in order 56 onstru ted from new ami able sets of eight matri es. 5 A1 A2 Type A3 A4 ZERO A5 A6 A7 A8 (1,14,14,27) ( a; d; d; d; d; d; d ) ( d; a; a; a; a; a; a ) PAF ( b; d; d; d; d; d; d ) ( d; b; b; b; b; b; b ) n=7 ( ; d; d; d; d; d; d ) ( d; d; d; d; d; d; d ) ( b; a; a; b; a; b; b ) ( a; b; b; a; b; a; a ) (1,20,35) ( a; a; a; a; a; a; a ) ( d; a; a; a; a; a; a ) PAF ( a; a; a; a; a; a; a ) ( a; b; b; a; b; a; a ) n=7 ( a; b; b; a; b; a; a ) ( a; a; a; a; a; a; a ) ( b; b; b; b; b; b; b ) ( b; b; b; b; b; b; b ) (2,2,8,8,18,18) ( a; b; b; a; b; a; a ) ( e; f; f; e; f; e; e ) PAF ( a; b; b; a; b; a; a ) ( e; f; f; e; f; e; e ) n=7 ( d; b; b; b; b; b; b ) ( d; b; b; b; b; b; b ) ( h; f; f; f; f; f; f ) ( h; f; f; f; f; f; f ) (2,4,22,28) ( a; a; a; a; a; a; a ) ( f; h; h; h; h; h; h ) PAF ( f; e; h; h; h; h; h ) ( a; a; a; a; a; a; a ) n=7 ( a; a; a; a; a; a; a ) ( f; e; h; h; h; h; h ) ( a; a; a; a; a; a; a ) ( f; h; h; h; h; h; h ) (3,22,31) ( a; b; b; a; b; a; a ) ( d; b; b; b; b; b; b ) PAF ( a; b; b; a; b; a; a ) ( d; b; b; b; b; b; b ) n=7 ( d; a; a; a; a; a; a ) ( a; b; b; b; b; b; b ) ( a; b; b; a; b; a; a ) ( b; a; a; b; a; b; b ) (4,4,4,4,10,10,10,10) ( b; ; a; ; d; d; d ) ( f; g; e; g; h; h; h ) NPAF ( b; ; a; ; d; d; d ) ( f; g; e; g; h; h; h ) n=7 ( b; d; a; d; ; ; ) ( f; h; e; h; g; g; g ) ( b; d; a; d; ; ; ) ( f; h; e; h; g; g; g ) (4,6,46) ( ; ; ; ; ; b; a ) ( ; ; ; ; a; b; ) PAF ( ; ; ; ; ; ; ) ( ; ; ; ; ; b; ) n=7 ( ; ; ; ; a; b; ) ( ; ; ; ; ; b; a ) ( ; ; ; ; ; b; ) ( ; ; ; ; ; ; ) (4,7,21,24) ( a; a; a; a; a; a; d ) ( f; f; f; f; e; f; f ) NPAF ( f; f; f; f; e; f; f ) ( a; a; a; a; d; a; a ) n=7 ( f; f; f; f; e; f; f ) ( a; a; a; d; a; d; a ) ( d; d; d; a; a; a; a ) ( f; f; f; f; e; f; f ) (7,7,7,7,7,7,7,7) ( a; a; a; g; a; e; ) ( f; f; f; h; f; b; d ) NPAF ( g; g; g; a; g; ; e ) ( h; h; h; f; h; d; b ) n=7 ( e; e; e; ; e; a; g ) ( d; d; d; b; d; h; f ) ( b; b; b; d; b; f; h ) ( ; ; ; e; ; g; a ) Table 3 ( ont.) 6 A1 A2 Type A3 A4 ZERO A5 A6 A7 A8 (7,7,18,24) ( a; a; a; a; ; a; d ) ( b; b; b; b; a; b; b ) NPAF ( b; b; b; b; a; b; b ) ( a; a; a; a; d; a; ) n=7 ( b; b; b; b; a; b; b ) ( ; ; ; d; a; d; a ) ( b; b; b; b; a; b; b ) ( d; d; d; ; a; ; a ) (8,11,37) ( a; b; b; a; b; a; a ) ( ; b; b; b; b; b; b) PAF ( a; b; b; a; b; a; a ) ( ; b; b; b; b; b; b ) n=7 ( b; b; b; b; b; b; b ) ( ; b; b; b; b; b; b ) ( ; b; b; ; b; ; ) ( ; b; b; ; b; ; ) (11,14,31) ( a; b; b; a; b; a; a ) ( b; a; a; b; ab; b ) PAF ( ; a; a; a; a; a; a ) ( a; b; b; b; b; b; b ) n=7 ( ; b; b; ; b; ; ) ( ; b; b; b; b; b; b ) ( ; b; b; ; b; ; ) ( ; b; b; b; b; b; b ) Table 3 ( ont.) Remark 1 We note that ami able sets of eight matri es of type (4; 4; 4; 4; 10; 10; 10; 10) and (7; 7; 7; 7; 7; 7; 7; 7) whi h are used for onstru ting OD's in order 56 are also found in [4℄. (1; 2; 3; 50) (1; 2; 25; 28) (1; 3; 8; 44) (1; 3; 13; 39) (1; 3; 14; 38) (1; 3; 19; 33) (1; 3; 25; 27) (1; 5; 25; 25) (1; 8; 19; 28) (1; 8; 22; 25) (1; 11; 19; 25) (1; 13; 14; 28) (1; 13; 17; 25) (1; 14; 16; 25) (2; 2; 8; 44) (2; 2; 16; 36) (2; 2; 18; 34) (2; 2; 26; 26) (2; 3; 25; 26) (2; 4; 25; 25) (2; 8; 8; 38) (2; 8; 10; 36) (2; 8; 18; 28) (2; 8; 20; 26) (2; 10; 18; 26) (2; 16; 18; 20) (2; 18; 18; 18) (3; 3; 25; 25) (3; 8; 19; 26) (3; 8; 20; 25) (3; 9; 19; 25) (3; 13; 14; 26) (3; 13; 15; 25) (3; 14; 14; 25) (4; 4; 4; 44) (4; 4; 8; 40) (4; 4; 10; 38) (4; 4; 14; 34) (4; 4; 18; 30) (4; 4; 20; 28) (4; 4; 24; 24) (4; 8; 8; 36) (4; 8; 10; 34) (4; 8; 14; 30) (4; 8; 18; 26) (4; 8; 19; 25) (4; 8; 20; 24) (4; 10; 10; 32) (4; 10; 12; 30) (4; 10; 14; 28) (4; 10; 18; 24) (4; 10; 20; 22) (4; 12; 20; 20) (4; 13; 14; 25) (4; 14; 14; 24) (4; 14; 18; 20) (4; 16; 18; 18) (7; 7; 7; 35) (7; 7; 14; 28) (7; 7; 21; 21) (7; 14; 14; 21) (8; 8; 10; 30) (8; 8; 18; 22) (8; 8; 20; 20) (8; 10; 10; 28) (8; 10; 14; 24) (8; 10; 18; 20) (8; 12; 18; 18) (8; 14; 14; 20) (10; 10; 10; 26) (10; 10; 12; 24) (10; 10; 14; 22) (10; 10; 16; 20) (10; 10; 18; 18) (10; 12; 14; 20) (10; 14; 14; 18) (14; 14; 14; 14) Table 4: Full 4-variable OD(56; s1; s2; s3; 56 s1 s2 s3) onstru ted from full designs presented in table 2. 3 Full designs with even parameters We note that Seberry [8℄ showed that if allOD(n;x; y; n x y) exist then allOD(2n; z; w; 2n z w) exist for s 0 an integer. In parti ular if all OD(2tp;x; y; 2tp x y) exist, for some odd integer p, then all OD(2t+sp; z; w; 2t+sp z w) exist for s 0 an integer we observe Lemma 2 If all OD(2tp; 2x; 2y; 2tp 2x 2y) exist, for some odd integer p, then all OD(2t+sp; 2z; 2w; 2t+sp 2z 2w) exist for s 0 an integer. 7 s1; s2; s3 (1; 9; 46) (1; 23; 32) (1; 24; 31) (2; 5; 49) (2; 7; 47) (2; 11; 43) (2; 23; 31) (3; 4; 49) (3; 6; 47) (3; 7; 46) (3; 10; 43) (3; 11; 42) (3; 21; 32) (3; 22; 31) s1; s2; s3 (3; 24; 29) (4; 5; 47) (4; 11; 41) (4; 15; 37) (4; 23; 29) (5; 6; 45) (5; 7; 44) (5; 8; 43) (5; 9; 42) (5; 11; 40) (5; 13; 38) (5; 14; 37) (5; 16; 35) (5; 17; 34) s1; s2; s3 (5; 19; 32) (5; 20; 31) (5; 21; 30) (5; 22; 29) (5; 24; 27) (6; 9; 41) (6; 11; 39) (6; 16; 34) (6; 17; 33) (6; 21; 29) (6; 23; 27) (7; 8; 41) (7; 9; 40) (7; 10; 39) s1; s2; s3 (7; 15; 34) (7; 16; 33) (7; 17; 32) (7; 19; 30) (7; 20; 29) (7; 22; 27) (7; 23; 26) (8; 9; 39) (8; 13; 35) (8; 15; 33) (8; 17; 31) (8; 21; 27) (9; 12; 35) (9; 14; 33) s1; s2; s3 (9; 15; 32) (9; 16; 31) (9; 17; 30) (9; 21; 26) (9; 23; 24) (10; 11; 35) (10; 13; 33) (10; 15; 31) (10; 17; 29) (10; 21; 25) (11; 12; 33) (11; 13; 32) (11; 15; 30) (11; 16; 29) s1; s2; s3 (11; 22; 23) (12; 13; 31) (12; 15; 29) (12; 17; 27) (13; 16; 27) (13; 19; 24) (13; 20; 23) (13; 21; 22) (15; 17; 24) (15; 19; 22) (16; 17; 23) (17; 19; 20) Table 5: The existen e of these 82 full OD(56; s1; s2; 72 s1 s2) is not yet established. Corollary 2 If OD(56; 6; 16; 34) exist then all OD(2s+37; 2z; 2w; 2s+37 2z 2w) exist for s 0 an integer. Proof. A sear h of full OD(56;x; y; 56 x y) show only the parameters indi ated are as yet unsolved. 2 4 Summary We have found new designs in order 56 and shown that of 1285 possible OD(56; s1; s2; s3; 56 s1 s2 s3) 163 are known: of 261 possible OD(56; s1; s2; 56 s1 s2) 179 are known; and all possible OD(56; s1; 56 s1) are known. Referen es [1℄ S. Georgiou, C. Koukouvinos, M. Mitrouli and J. Seberry, Ne essary and suÆ ient onditions for three and four variable orthogonal designs in order 36, J. Statist. Plann. Inferen e, (to appear). [2℄ A.V.Geramita, J.M.Geramita, and J.Seberry Wallis, Orthogonal designs, Linear and Multilinear Algebra, 3 (1976), 281-306. [3℄ A.V.Geramita, and J.Seberry, Orthogonal Designs: Quadrati Forms and Hadamard Matri es, Mar el Dekker, New York-Basel, 1979. [4℄ W.H. Holzmann, and H. Kharaghani, On the Plotkin arrays, Australas. J. Combin., to appear. [5℄ C.Koukouvinos, M.Mitrouli, J.Seberry, and P.Karabelas, On suÆ ient onditions for some orthogonal designs and sequen es with zero auto orrelation fun tion, Australas. J. Combin., 13 (1996), 197-216. [6℄ C. Koukouvinos and J. Seberry, New orhogonal designs and sequen es with two and three variables in order 28, Ars Combinatoria, 54 (2000), 97-108. 8 [7℄ H. Kharaghani, Arrays for orthogonal designs, J. Combin. Designs, to appear. [8℄ J. Seberry Wallis, On the existen e of Hadamard matri es, J. Combin. Theory Ser. A, 21 (1976), 188-195.