The Gale-Berlekamp game for Hadamard matrices

Given an Hadamard matrix $H\in M_N(\pm 1)$ we consider the function $\phi :{\mathbb{Z}}_2^N\times {\mathbb{Z}}_2^N\to \mathbb{Z}$ given by $\phi (a,b)={\sum }_{ij}{a}_i{b}_j{H}_{ij}$, which sums the entries of the various conjugates of $H$, obtained by switching signs on rows and columns. Our claim is that $\phi$, 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 $\mu$.