Journal of Algebraic Combinatorics

, Volume 42, Issue 1, pp 283–291 | Cite as

Weighing matrices and spherical codes

Article

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.

Keywords

Weighing matrices Mutually unbiased weighing matrices  Root system Kerdock codes over \(\mathbb {Z}_{4}\) 

Notes

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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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Mathematics EducationAichi University of EducationKariyaJapan

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