{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:GHQ5434ZK4M2HYQQHDOGXQVAUC","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":"54bf8d08cabb4413558ba4e9c67426e290d6fb10b47b37556049cd71916ee16a","cross_cats_sorted":["cs.IT","math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2023-11-03T14:38:05Z","title_canon_sha256":"de2e81d989cff01b9c92d52d1ff0548ff45009a8e43cd4ab98fc978288788ecb"},"schema_version":"1.0","source":{"id":"2311.01947","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2311.01947","created_at":"2026-07-05T10:15:41Z"},{"alias_kind":"arxiv_version","alias_value":"2311.01947v2","created_at":"2026-07-05T10:15:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2311.01947","created_at":"2026-07-05T10:15:41Z"},{"alias_kind":"pith_short_12","alias_value":"GHQ5434ZK4M2","created_at":"2026-07-05T10:15:41Z"},{"alias_kind":"pith_short_16","alias_value":"GHQ5434ZK4M2HYQQ","created_at":"2026-07-05T10:15:41Z"},{"alias_kind":"pith_short_8","alias_value":"GHQ5434Z","created_at":"2026-07-05T10:15:41Z"}],"graph_snapshots":[{"event_id":"sha256:79ad8159c19f4c170daba7804a2a4a74905aed2626be2ee34bf1d829167c52d9","target":"graph","created_at":"2026-07-05T10:15:41Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2311.01947/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A linear code $C$ over $\\mathbb{F}_q$ is called $\\Delta$-divisible if the Hamming weights $\\operatorname{wt}(c)$ of all codewords $c \\in C$ are divisible by $\\Delta$. The possible effective lengths of $q^r$-divisible codes have been completely characterized for each prime power $q$ and each non-negative integer $r$. The study of $\\Delta$ divisible codes was initiated by Harold Ward. If $c$ divides $\\Delta$ but is coprime to $q$, then each $\\Delta$-divisible code $C$ over $\\F_q$ is the $c$-fold repetition of a $\\Delta/c$-divisible code. Here we determine the possible effective lengths of $p^r$-","authors_text":"Sascha Kurz","cross_cats":["cs.IT","math.IT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2023-11-03T14:38:05Z","title":"Lengths of divisible codes -- the missing cases"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2311.01947","kind":"arxiv","version":2},"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":"sha256:9eade5f364273b8dfe66236536c0eac7c9fd570c5f70c8fc454dcdb140dcaf6f","target":"record","created_at":"2026-07-05T10:15:41Z","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":"54bf8d08cabb4413558ba4e9c67426e290d6fb10b47b37556049cd71916ee16a","cross_cats_sorted":["cs.IT","math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2023-11-03T14:38:05Z","title_canon_sha256":"de2e81d989cff01b9c92d52d1ff0548ff45009a8e43cd4ab98fc978288788ecb"},"schema_version":"1.0","source":{"id":"2311.01947","kind":"arxiv","version":2}},"canonical_sha256":"31e1de6f995719a3e21038dc6bc2a0a08b79df476f30fb78b2596c6a84b84deb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"31e1de6f995719a3e21038dc6bc2a0a08b79df476f30fb78b2596c6a84b84deb","first_computed_at":"2026-07-05T10:15:41.807578Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T10:15:41.807578Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3dfwgT5OWzSOkmdMZICyjViBXNnQ79jOCpIyGM2r17IsmKjhW5g2RUNco3g3Fv7wPP7nFUDtvr3q+moHMqnOCA==","signature_status":"signed_v1","signed_at":"2026-07-05T10:15:41.808057Z","signed_message":"canonical_sha256_bytes"},"source_id":"2311.01947","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9eade5f364273b8dfe66236536c0eac7c9fd570c5f70c8fc454dcdb140dcaf6f","sha256:79ad8159c19f4c170daba7804a2a4a74905aed2626be2ee34bf1d829167c52d9"],"state_sha256":"0acd9db654f310aa79e4490c2bb1deb09418059a1bcf131edf15a95ddb64c7a6"}