Regularity questions for complex Hadamard matrices

TL;DR

This paper investigates regular complex Hadamard matrices, focusing on the case M=3, providing classifications at N=7, and exploring potential applications of these results to broader matrix questions.

Contribution

The paper introduces a classification of regular complex Hadamard matrices for M=3 and N=7, and discusses their potential applications to other matrix problems.

Findings

Classification of regular complex Hadamard matrices at N=7 for M=3

Analysis of scalar products decomposing as sums of cycles

Discussion of applications to M=N Hadamard matrix questions

Abstract

We study the partial Hadamard matrices $H\in M_{M\times N}(\mathbb C)$ which are regular, in the sense that the scalar products between pairs of distinct rows decompose as sums of cycles (rotated sums of roots of unity). The simplest non-trivial case is M=3, and we obtain here several results, notably with a classification at N=7. We discuss as well the potential applications of the M=3 results to various $M=N$ questions.