Journal of Algebraic Combinatorics

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

Weighing matrices and spherical codes



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.


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



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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© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Mathematics EducationAichi University of EducationKariyaJapan

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