The Gale-Berlekamp game for Hadamard matrices

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 $\mu$.