RIS ID

7786

Publication Details

This article was originally published as Georgiou, S, Koukouvinos, C, Mitrouli, M and Seberry, J, Necessary and Sufficient Conditions for Three and Four Variable Orthogonal Designs in Order 36, Journal of Statistical Planning and Inference, 106, 2002, 329-352. Copyright Elsevier. Original journal available here.

Abstract

We use a new algorithm to find new sets of sequences with entries from {0, ±a, ±b, ±c, ±d}, on the commuting variables a, b, c, d, with zero autocorrelation function. Then we use these sequences to construct a series of new three and four variable orthogonal designs in order 36. We show that the necessary conditions plus (.s1, s2, s3, s4) not equal to 12816 18 816 221313 26721 36 816 4889 12825 191313 23 424 289 9 381015 8899 14425 22 916 are sufficient for the existence of an OD(36; s1, s2 s3, s4) constructed using four circulant matrices in the Goethals-Seidel array. Of the 154 theoretically possible cases 133 are known. We also show that the necessary conditions plus (s1, s2 s3) ≠ (2,8,25), (6,7,21), (8, 9, 17) or (9, 13, 13) are sufficient for the existence of an OD(36; s1, s2, s3) constructed using four circulant matrices in the Goethals-Seidel array. Of the 433 theoretically possible cases 429 are known. Further, we show that the necessary conditions are sufficient for the existence of an OD(36; s1, s2 36 — s1 — s2) in each of the 54 theoretically possible cases. Further, of the 27 theoretically possible OD(36; s1, s2, s3, 36 — s1 — s2 — s3), 23 are known to exist, and four, (1,2,8,25), (1, 9,13,13), (2,6,7,21) and (3, 8,10,15), cannot be constructed using four circulant matrices. By suitably replacing the variables by ±1 these lead to more than 200 potentially inequivalent Hadamard matrices of order 36. By considering the 12 OD(36; 1, s1, 35—s1) and suitably replacing the variables by ±1 we obtain 48 potentially inequivalent skew-Hadamard matrices of order 36. A summary with all known results in order 36 is presented in the Tables.

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Link to publisher version (DOI)

http://dx.doi.org/10.1016/S0378-3758(02)00221-5