The Gale-Berlekamp game for Hadamard matrices

Given an Hadamard matrix $H\in M_N(\pm1)$ we consider the function $φ:\mathbb Z_2^N\times\mathbb Z_2^N\to\mathbb Z$ given by $φ(a,b)=\sum_{ij}a_ib_jH_{ij}$, which sums the entries of the various conjugates of $H$, obtained by switching signs on rows and columns. Our claim is that $φ$, or just its probabilistic distribution $μ\in\mathcal P(\mathbb Z)$, 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 $μ$.

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