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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1806.04988","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-06-13T12:43:19Z","cross_cats_sorted":[],"title_canon_sha256":"5f73c35c125b70aab568365658c2af2067dc1ae629a1585469d4d368243e10bd","abstract_canon_sha256":"91c4302a51223d4788d4caf7289a67bc55bb40312d5d7078a809a17fcbba52f8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:00:39.499846Z","signature_b64":"wgo9qQRf1ztFJU/YG/JYOnInohGENg2g58lWi9ILzvQ9dt41bHRio1SEHScrCQDY3VfrxYFIzkCL2tQWeqv7Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c6bbc8e75cb3a11c5aea111ffeea9b11a34af32dd1820dfcfe1b4bdc3627a099","last_reissued_at":"2026-05-18T00:00:39.499279Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:00:39.499279Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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