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A sensitive algorithm for detecting the inequivalence of Hadamard matrices

Authors: Kai-Tai Fang and Gennian Ge
Journal: Math. Comp. 73 (2004), 843-851
MSC (2000): Primary 68Q15, 05B20, 62K15
Published electronically: September 2, 2003
MathSciNet review: 2031409
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Abstract: A Hadamard matrix of side $n$ is an $n \times n$ matrix with every entry either $1$ or $-1$, which satisfies $HH^{T}=nI$. Two Hadamard matrices are called equivalent if one can be obtained from the other by some sequence of row and column permutations and negations. To identify the equivalence of two Hadamard matrices by a complete search is known to be an NP hard problem when $n$increases. In this paper, a new algorithm for detecting inequivalence of two Hadamard matrices is proposed, which is more sensitive than those known in the literature and which has a close relation with several measures of uniformity. As an application, we apply the new algorithm to verify the inequivalence of the known $60$ inequivalent Hadamard matrices of order $24$; furthermore, we show that there are at least $382$ pairwise inequivalent Hadamard matrices of order $36$. The latter is a new discovery.

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Additional Information

Kai-Tai Fang
Affiliation: Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China

Gennian Ge
Affiliation: Department of Mathematics, Suzhou University, Suzhou, 215006, China
Address at time of publication: Department of Mathematics, Zhejiang University, Hongzhou 310027, Zhejiang, China

Keywords: Algorithm, equivalence, Hadamard matrix, Hamming distance, uniformity
Received by editor(s): July 9, 2001
Received by editor(s) in revised form: November 28, 2001
Published electronically: September 2, 2003
Additional Notes: This research was supported in part by the Hong Kong RGC grants HKBU RC/98-99/Gen-370 and HKBU 2044/02P. The second author was also supported by statistics Research and Consultancy Centre, Hong Kong Baptist University and the YNSFC Grant 10001026
Article copyright: © Copyright 2003 American Mathematical Society