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

TL;DR

This paper explores quasi-unbiased and weakly unbiased Hadamard matrices using coding theory, providing bounds, classifications, and modifications to their definitions, advancing understanding of their structure and relationships.

Contribution

It introduces a coding-theoretic framework to study these matrices, establishes upper bounds, classifies certain codes, and proposes modifications to the weakly unbiased concept.

Findings

Upper bounds on the number of mutually quasi-unbiased Hadamard matrices.

Classifications of self-complementary codes related to these matrices.

Proposed modifications to the definition of weakly unbiased Hadamard matrices.

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.