Some special complex Hadamard matrices of order six

TL;DR

This paper advances the classification of 6x6 complex Hadamard matrices by characterizing specific subclasses with unique structural properties, contributing to the longstanding open problem in matrix theory.

Contribution

It provides a detailed characterization of several special classes of 6x6 complex Hadamard matrices, marking progress toward their complete classification.

Findings

Characterized CHMs with a 2x3 rank-one submatrix

Identified CHMs with three distinct elements and first row all ones

Analyzed H2-reducible CHMs with three distinct elements

Abstract

The complete classification of $6\times 6$ complex Hadamard matrices (CHMs) is a long-standing open problem. In this paper we investigate a series of CHMs, such as the CHMs containing a $2\times 3$ submatrix with rank one, the CHMs containing exactly three distinct elements and all elements of the first row being one, the $H_2$-reducible matrices containing exactly three distinct matrix elements, and the CHMs containing a $3\times 3$ Hadamard submatrix. We characterize all forms of these CHMs. Our results are the latest progress on the complete classification of CHMs.