{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/fooling-sets-and-rank","title":"Fooling sets and rank","arxiv_id":"1208.2920","date":"2012-08-14","proceeding":null,"authors":["Mirjam Friesen","Aya Hamed","Troy Lee","Dirk Oliver Theis"],"abstract":"An $n\\times n$ matrix $M$ is called a \\textit{fooling-set matrix of size $n$} if its diagonal entries are nonzero and $M_{k,\\ell} M_{\\ell,k} = 0$ for every $k\\ne \\ell$. Dietzfelbinger, Hromkovi{\\v{c}}, and Schnitger (1996) showed that $n \\le (\\mbox{rk} M)^2$, regardless of over which field the rank is computed, and asked whether the exponent on $\\mbox{rk} M$ can be improved. We settle this question. In characteristic zero, we construct an infinite family of rational fooling-set matrices with size $n = \\binom{\\mbox{rk} M+1}{2}$. In nonzero characteristic, we construct an infinite family of matrices with $n= (1+o(1))(\\mbox{rk} M)^2$.","url_abs":"http://arxiv.org/abs/1208.2920v3","url_pdf":"http://arxiv.org/pdf/1208.2920v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"fooling-sets-and-rank","repo_url":"https://github.com/maehjam/maehjam.github.io","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}