The Gale-Berlekamp game for Hadamard matrices

TL;DR

This paper explores the 'glow' distribution derived from Hadamard matrices, proposing it encodes significant structural information and presenting results and conjectures about its properties.

Contribution

It introduces the concept of the glow distribution of Hadamard matrices and provides a decomposition result, offering new insights into their structure.

Findings

Decomposition result for the glow distribution

Conjectures relating glow to Hadamard matrix properties

Potential encoding of structural information in glow

Abstract

Given an Hadamard matrix $H\in M_N(\pm1)$ we consider the function $\varphi:\mathbb Z_2^N\times\mathbb Z_2^N\to\mathbb Z$ given by $\varphi(a,b)=\sum_{ij}a_ib_jH_{ij}$, which sums the entries of the various conjugates of $H$, obtained by switching signs on rows and columns. Our claim is that $\varphi$, or just its probabilistic distribution $\mu\in\mathcal P(\mathbb Z)$, that we call "glow" of the matrix, should encode important information about $H$. We present here a number of results and conjectures in this direction, notably with a general decomposition result for $\mu$.