{"paper":{"title":"On the rank of a random binary matrix","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"A.M. Frieze, C. Cooper, W.Pegden","submitted_at":"2018-06-13T12:43:19Z","abstract_excerpt":"We study the rank of the random $n\\times m$ 0/1 matrix ${\\bf A}_{n,m;k}$ where each column is chosen independently from the set $\\Omega_{n,k}$ of 0/1 vectors with exactly $k$ 1's. Here 0/1 are the elements of the field $GF_2$. We obtain an asymptotically correct estimate for the rank in terms of $c,n,k$, assuming that $m=cn$.\n  In addition, we assign i.i.d. $U[0,1]$ weights $X_{{\\bf c}},{\\bf c}\\in\\Omega_{n,k}$ and let the weight of a set of columns $C$ be $X(C)=\\sum_{{\\bf c}\\in C}X_{{\\bf c}}$. Let a basis be a set of $n-1_{k\\text{even}}$ linearly independent columns. We obtain an asymptoticall"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.04988","kind":"arxiv","version":2},"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"}