{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,7]],"date-time":"2026-03-07T18:08:10Z","timestamp":1772906890635,"version":"3.50.1"},"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>We study the rank of a random $n \\times m$ matrix $\\mathbf{A}_{n,m;k}$ with entries from $GF(2)$, and exactly $k$ unit entries in each column, the other entries being zero. The columns are chosen independently and uniformly at random from the set of all ${n \\choose k}$ such columns.\r\nWe obtain an asymptotically correct estimate for the rank as a function of the number of columns $m$ in terms of $c,n,k$, and where $m=cn\/k$. The matrix $\\mathbf{A}_{n,m;k}$ forms the vertex-edge incidence matrix of a $k$-uniform random hypergraph $H$. The rank of $\\mathbf{A}_{n,m;k}$ can be expressed as follows. Let $|C_2|$ be the number of vertices of the 2-core of $H$, and $|E(C_2)|$ the number of edges. Let $m^*$ be the value of $m$ for which $|C_2|= |E(C_2)|$. Then w.h.p. for $m&lt;m^*$ the rank of $\\mathbf{A}_{n,m;k}$ is asymptotic to $m$, and for $m \\ge m^*$ the rank is asymptotic to $m-|E(C_2)|+|C_2|$.\r\nIn addition, assign i.i.d. $U[0,1]$ weights $X_i, i \\in {1,2,...m}$ to the columns, and define the weight of a set of columns $S$ as $X(S)=\\sum_{j \\in S} X_j$. Define a basis as a set of $n-\ud835\udfd9 (k\\text{ even})$ linearly independent columns. We obtain an asymptotically correct estimate for the minimum weight basis. This generalises the well-known result of Frieze [On the value of a random minimum spanning tree problem, Discrete Applied Mathematics, (1985)] that, for $k=2$, \u00a0 the expected length of a minimum weight spanning tree tends to $\\zeta(3)\\sim 1.202$.<\/jats:p>","DOI":"10.37236\/8092","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T06:04:58Z","timestamp":1578636298000},"source":"Crossref","is-referenced-by-count":6,"title":["On the Rank of a Random Binary Matrix"],"prefix":"10.37236","volume":"26","author":[{"given":"Colin","family":"Cooper","sequence":"first","affiliation":[]},{"given":"Alan","family":"Frieze","sequence":"additional","affiliation":[]},{"given":"Wesley","family":"Pegden","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2019,10,11]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v26i4p12\/7937","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v26i4p12\/7937","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T04:02:50Z","timestamp":1579233770000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v26i4p12"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,10,11]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2019,10,11]]}},"URL":"https:\/\/doi.org\/10.37236\/8092","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,10,11]]},"article-number":"P4.12"}}