Necessary and sufficient conditions for three and four variable orthogonal designs in order 36

Georgiou, S; Koukouvinos, C; Mitrouli, M; Seberry, J

dc.contributor.author	Georgiou, S	en
dc.contributor.author	Koukouvinos, C	en
dc.contributor.author	Mitrouli, M	en
dc.contributor.author	Seberry, J	en
dc.date.accessioned	2014-03-01T01:18:05Z
dc.date.available	2014-03-01T01:18:05Z
dc.date.issued	2002	en
dc.identifier.issn	0378-3758	en
dc.identifier.uri	https://dspace.lib.ntua.gr/xmlui/handle/123456789/14793
dc.subject	Algorithm	en
dc.subject	Autocorrelation	en
dc.subject	Circulant matrix	en
dc.subject	Construction	en
dc.subject	Orthogonal design	en
dc.subject.classification	Statistics & Probability	en
dc.title	Necessary and sufficient conditions for three and four variable orthogonal designs in order 36	en
heal.type	journalArticle	en
heal.identifier.primary	10.1016/S0378-3758(02)00221-5	en
heal.identifier.secondary	http://dx.doi.org/10.1016/S0378-3758(02)00221-5	en
heal.language	English	en
heal.publicationDate	2002	en
heal.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 1 2 8 16 1 8 8 16 2 2 13 13 2 6 7 21 3 6 8 4 8 8 9 1 2 8 25 1 9 13 13 2 3 4 24 2 8 9 9 3 8 10 15 8 8 9 9 1 4 4 25 2 2 9 16 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. © 2002 Elsevier Science B.V. All rights reserved.	en
heal.publisher	ELSEVIER SCIENCE BV	en
heal.journalName	Journal of Statistical Planning and Inference	en
dc.identifier.doi	10.1016/S0378-3758(02)00221-5	en
dc.identifier.isi	ISI:000177649000027	en
dc.identifier.volume	106	en
dc.identifier.issue	1-2	en
dc.identifier.spage	329	en
dc.identifier.epage	352	en