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Weighing matrices and spherical codes

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Abstract

Mutually unbiased weighing matrices (MUWM) are closely related to an antipodal spherical code with 4 angles. In this paper, we clarify the relation between MUWM and the spherical codes and determine the maximum size of a set of MUWM with weight 4 for any order. Moreover, we define mutually quasi-unbiased weighing matrices (MQUWM) as a natural generalization of MUWM from the viewpoint of spherical codes. We determine the maximum size of a set of MQUWM for the parameters \((d,2,4,1)\) and \((d,d,d/2,2d)\). This includes an affirmative answer to the problem of Best, Kharaghani, and Ramp.

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Acknowledgments

We would like to thank the two anonymous refrees for their valuable comments for the first version of this paper. The first author was supported by JSPS KAKENHI Grant Number 25800011.

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  1. Department of Mathematics Education, Aichi University of Education, 1 Hirosawa, Igaya-cho, Kariya, Aichi, 448-8542, Japan

    Hiroshi Nozaki & Sho Suda

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  1. Hiroshi Nozaki
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  2. Sho Suda
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Correspondence to Sho Suda.

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Nozaki, H., Suda, S. Weighing matrices and spherical codes. J Algebr Comb 42, 283–291 (2015). https://doi.org/10.1007/s10801-015-0581-6

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  • DOI: https://doi.org/10.1007/s10801-015-0581-6

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