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

DOI:
https://doi.org/10.1090/S0025-5718-03-01539-4

Published electronically:
September 2, 2003

MathSciNet review:
2031409

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A Hadamard matrix of side is an matrix with every entry either or , which satisfies . 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 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 inequivalent Hadamard matrices of order ; furthermore, we show that there are at least pairwise inequivalent Hadamard matrices of order . 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

Email:
ktfang@math.hkbu.edu.hk

**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

Email:
gnge@public1.sz.js.cn

DOI:
https://doi.org/10.1090/S0025-5718-03-01539-4

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