Weighing matrices and spherical codes
Hiroshi Nozaki, Sho Suda
TL;DR
This paper explores the relationship between mutually unbiased weighing matrices and spherical codes, providing a complete solution for their maximum size and generalizations, advancing understanding in combinatorial design theory.
Contribution
It clarifies the connection between MUWM and spherical codes and determines maximum sizes for MUWM and their generalizations, solving an open problem.
Findings
Complete solution for maximum size of MUWM of weight 4
Determined maximum sizes of generalized MUWM sets
Confirmed an open problem by Best, Kharaghani, and Ramp
Abstract
Mutually unbiased weighing matrices (MUWM) are closely related to an antipodal spherical code with 4 angles. In the present paper, we clarify the relationship between MUWM and the spherical sets, and give the complete solution about the maximum size of a set of MUWM of weight 4 for any order. Moreover we describe some natural generalization of a set of MUWM from the viewpoint of spherical codes, and determine several maximum sizes of the generalized sets. They include an affirmative answer of the problem of Best, Kharaghani, and Ramp.
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Taxonomy
Topicsgraph theory and CDMA systems · Coding theory and cryptography · Finite Group Theory Research