{"paper":{"title":"The Gale-Berlekamp game for Hadamard matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Teodor Banica","submitted_at":"2013-06-25T15:37:32Z","abstract_excerpt":"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 $\\mu$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.6003","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}