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We say that $\\mathcal{C}$ is asymptotically good if, for some $\\varepsilon > 0$ and for all $i$, $n_i \\geq i$, $k_i/n_i \\geq \\varepsilon$, and $d_i/n_i \\geq \\varepsilon$. Sequences of asymptotically good codes exist. We prove that if $\\mathcal{C}$ is a class of GF$(p^n)$-linear codes (where $p$ is prime and $n \\geq 1$), closed under pun"},"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":"1404.7771","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-04-30T15:59:02Z","cross_cats_sorted":[],"title_canon_sha256":"f0408f97d5be9fe86dccb988186aa34331b608e4e0e2a3e2f48c4245b8526208","abstract_canon_sha256":"4f565d29b1f5b46ba64aeea31b0ca9ce9c1ecc20ef8bc60d3c97f41aefaebc50"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:52:49.993451Z","signature_b64":"Qrx7yLDzm9KhmMQ3WNACzqccLnYCN54qYee1XAhCSabb9IV9VXKrsS++An+bCVQHuf5YFUemZFFqOJY84BIADw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3c5036b15ab784545ee1a1a202684bb676ec1c0070573cc063c776ac95590857","last_reissued_at":"2026-05-18T02:52:49.992957Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:52:49.992957Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the existence of asymptotically good linear codes in minor-closed classes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Peter Nelson, Stefan H.M. van Zwam","submitted_at":"2014-04-30T15:59:02Z","abstract_excerpt":"Let $\\mathcal{C} = (C_1, C_2, \\ldots)$ be a sequence of codes such that each $C_i$ is a linear $[n_i,k_i,d_i]$-code over some fixed finite field $\\mathbb{F}$, where $n_i$ is the length of the codewords, $k_i$ is the dimension, and $d_i$ is the minimum distance. We say that $\\mathcal{C}$ is asymptotically good if, for some $\\varepsilon > 0$ and for all $i$, $n_i \\geq i$, $k_i/n_i \\geq \\varepsilon$, and $d_i/n_i \\geq \\varepsilon$. Sequences of asymptotically good codes exist. We prove that if $\\mathcal{C}$ is a class of GF$(p^n)$-linear codes (where $p$ is prime and $n \\geq 1$), closed under pun"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.7771","kind":"arxiv","version":1},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1404.7771","created_at":"2026-05-18T02:52:49.993027+00:00"},{"alias_kind":"arxiv_version","alias_value":"1404.7771v1","created_at":"2026-05-18T02:52:49.993027+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1404.7771","created_at":"2026-05-18T02:52:49.993027+00:00"},{"alias_kind":"pith_short_12","alias_value":"HRIDNMK2W6CF","created_at":"2026-05-18T12:28:30.664211+00:00"},{"alias_kind":"pith_short_16","alias_value":"HRIDNMK2W6CFIXXB","created_at":"2026-05-18T12:28:30.664211+00:00"},{"alias_kind":"pith_short_8","alias_value":"HRIDNMK2","created_at":"2026-05-18T12:28:30.664211+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/HRIDNMK2W6CFIXXBUGRAE2CLWZ","json":"https://pith.science/pith/HRIDNMK2W6CFIXXBUGRAE2CLWZ.json","graph_json":"https://pith.science/api/pith-number/HRIDNMK2W6CFIXXBUGRAE2CLWZ/graph.json","events_json":"https://pith.science/api/pith-number/HRIDNMK2W6CFIXXBUGRAE2CLWZ/events.json","paper":"https://pith.science/paper/HRIDNMK2"},"agent_actions":{"view_html":"https://pith.science/pith/HRIDNMK2W6CFIXXBUGRAE2CLWZ","download_json":"https://pith.science/pith/HRIDNMK2W6CFIXXBUGRAE2CLWZ.json","view_paper":"https://pith.science/paper/HRIDNMK2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1404.7771&json=true","fetch_graph":"https://pith.science/api/pith-number/HRIDNMK2W6CFIXXBUGRAE2CLWZ/graph.json","fetch_events":"https://pith.science/api/pith-number/HRIDNMK2W6CFIXXBUGRAE2CLWZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HRIDNMK2W6CFIXXBUGRAE2CLWZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HRIDNMK2W6CFIXXBUGRAE2CLWZ/action/storage_attestation","attest_author":"https://pith.science/pith/HRIDNMK2W6CFIXXBUGRAE2CLWZ/action/author_attestation","sign_citation":"https://pith.science/pith/HRIDNMK2W6CFIXXBUGRAE2CLWZ/action/citation_signature","submit_replication":"https://pith.science/pith/HRIDNMK2W6CFIXXBUGRAE2CLWZ/action/replication_record"}},"created_at":"2026-05-18T02:52:49.993027+00:00","updated_at":"2026-05-18T02:52:49.993027+00:00"}