{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:HRIDNMK2W6CFIXXBUGRAE2CLWZ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"4f565d29b1f5b46ba64aeea31b0ca9ce9c1ecc20ef8bc60d3c97f41aefaebc50","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-04-30T15:59:02Z","title_canon_sha256":"f0408f97d5be9fe86dccb988186aa34331b608e4e0e2a3e2f48c4245b8526208"},"schema_version":"1.0","source":{"id":"1404.7771","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1404.7771","created_at":"2026-05-18T02:52:49Z"},{"alias_kind":"arxiv_version","alias_value":"1404.7771v1","created_at":"2026-05-18T02:52:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1404.7771","created_at":"2026-05-18T02:52:49Z"},{"alias_kind":"pith_short_12","alias_value":"HRIDNMK2W6CF","created_at":"2026-05-18T12:28:30Z"},{"alias_kind":"pith_short_16","alias_value":"HRIDNMK2W6CFIXXB","created_at":"2026-05-18T12:28:30Z"},{"alias_kind":"pith_short_8","alias_value":"HRIDNMK2","created_at":"2026-05-18T12:28:30Z"}],"graph_snapshots":[{"event_id":"sha256:0ae9535156d82da3253ac12204688ab38e29f404cbabbd28abd1f6462b80d61e","target":"graph","created_at":"2026-05-18T02:52:49Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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","authors_text":"Peter Nelson, Stefan H.M. van Zwam","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-04-30T15:59:02Z","title":"On the existence of asymptotically good linear codes in minor-closed classes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.7771","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:8c63f42e4265c9f8d4738c991a634fc225f5e3fdb69fce44459a3f2956af1df2","target":"record","created_at":"2026-05-18T02:52:49Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"4f565d29b1f5b46ba64aeea31b0ca9ce9c1ecc20ef8bc60d3c97f41aefaebc50","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-04-30T15:59:02Z","title_canon_sha256":"f0408f97d5be9fe86dccb988186aa34331b608e4e0e2a3e2f48c4245b8526208"},"schema_version":"1.0","source":{"id":"1404.7771","kind":"arxiv","version":1}},"canonical_sha256":"3c5036b15ab784545ee1a1a202684bb676ec1c0070573cc063c776ac95590857","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3c5036b15ab784545ee1a1a202684bb676ec1c0070573cc063c776ac95590857","first_computed_at":"2026-05-18T02:52:49.992957Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:52:49.992957Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Qrx7yLDzm9KhmMQ3WNACzqccLnYCN54qYee1XAhCSabb9IV9VXKrsS++An+bCVQHuf5YFUemZFFqOJY84BIADw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:52:49.993451Z","signed_message":"canonical_sha256_bytes"},"source_id":"1404.7771","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8c63f42e4265c9f8d4738c991a634fc225f5e3fdb69fce44459a3f2956af1df2","sha256:0ae9535156d82da3253ac12204688ab38e29f404cbabbd28abd1f6462b80d61e"],"state_sha256":"5123876244b1bbec1ee5e11a853359635cb4133c1d71f2a794d04249d30f9eb0"}