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Quasi-unbiased Hadamard matrices and weakly unbiased Hadamard matrices: A coding-theoretic approach


Authors: Makoto Araya, Masaaki Harada and Sho Suda
Journal: Math. Comp.
MSC (2010): Primary 05B20, 94B25, 94B65; Secondary 05E30
Published electronically: June 29, 2016
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Abstract: This paper is concerned with quasi-unbiased Hadamard matrices and weakly unbiased Hadamard matrices, which are generalizations of unbiased Hadamard matrices, equivalently unbiased bases. These matrices are studied from the viewpoint of coding theory. As a consequence of a coding-theoretic approach, we provide upper bounds on the number of mutually quasi-unbiased Hadamard matrices. We give classifications of a certain class of self-complementary codes for modest lengths. These codes give quasi-unbiased Hadamard matrices and weakly unbiased Hadamard matrices. Some modification of the notion of weakly unbiased Hadamard matrices is also provided.


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Additional Information

Makoto Araya
Affiliation: Department of Computer Science, Shizuoka University, Hamamatsu 432–8011, Japan
Email: araya@inf.shizuoka.ac.jp

Masaaki Harada
Affiliation: Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai 980–8579, Japan
Email: mharada@m.tohoku.ac.jp

Sho Suda
Affiliation: Department of Mathematics Education, Aichi University of Education, Kariya 448-8542, Japan
Email: suda@auecc.aichi-edu.ac.jp

DOI: https://doi.org/10.1090/mcom/3122
Keywords: Unbiased Hadamard matrix, unbiased weighing matrix, self-complementary code
Received by editor(s): April 6, 2015
Received by editor(s) in revised form: September 16, 2015, and September 30, 2015
Published electronically: June 29, 2016
Dedicated: Dedicated to Professor Satoshi Yoshiara on his 60th birthday
Article copyright: © Copyright 2016 American Mathematical Society