{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:34Z","timestamp":1753893814784,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Let $\\mathbb{F}_{q}$ be a finite field of order $q$ with characteristic $p$. An arc is an ordered family of at least $k$ vectors in $\\mathbb{F}_{q}^{k}$ in which every subfamily of size $k$ is a basis of $\\mathbb{F}_{q}^{k}$. The MDS conjecture, which was posed by Segre in 1955, states that if $k \\leq q$, then an arc in $\\mathbb{F}_{q}^{k}$ has size at most $q+1$, unless $q$ is even and $k=3$ or $k=q-1$, in which case it has size at most $q+2$. We propose a conjecture which would imply that the MDS conjecture is true for almost all values of $k$ when $q$ is odd. We prove our conjecture in two cases and thus give simpler proofs of the MDS conjecture when $k \\leq p$, and if $q$ is not prime, for $k \\leq 2p-2$. To accomplish this, given an arc $G \\subset \\mathbb{F}_{q}^{k}$ and a nonnegative integer $n$, we construct a matrix $M_{G}^{\\uparrow n}$, which is related to an inclusion matrix, a well-studied object in combinatorics. Our main results relate algebraic properties of the matrix $M_{G}^{\\uparrow n}$ to properties of the arc $G$ and may provide new tools in the computational classification of large arcs.<\/jats:p>","DOI":"10.37236\/5713","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T19:52:44Z","timestamp":1578685964000},"source":"Crossref","is-referenced-by-count":4,"title":["Inclusion Matrices and the MDS Conjecture"],"prefix":"10.37236","volume":"23","author":[{"given":"Ameera","family":"Chowdhury","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2016,11,25]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v23i4p29\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v23i4p29\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T05:10:32Z","timestamp":1579237832000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v23i4p29"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,11,25]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2016,10,14]]}},"URL":"https:\/\/doi.org\/10.37236\/5713","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2016,11,25]]},"article-number":"P4.29"}}