{"paper":{"title":"Fooling-sets and rank in nonzero characteristic (extended abstract)","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dirk Oliver Theis, Mirjam Friesen","submitted_at":"2013-05-02T07:28:13Z","abstract_excerpt":"An n\\times n matrix M is called a fooling-set matrix of size n, if its diagonal entries are nonzero, whereas for every k\\ne \\ell we have M_{k,\\ell} M_{\\ell,k} = 0. Dietzfelbinger, Hromkovi\\v{c}, and Schnitger (1996) showed that n \\le (\\rk M)^2, regardless of over which field the rank is computed, and asked whether the exponent on \\rk M can be improved.\n  We settle this question for nonzero characteristic by constructing a family of matrices for which the bound is asymptotically tight. The construction uses linear recurring sequences."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.2468","kind":"arxiv","version":1},"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"}