{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:UMAVNRWLPX5E3SE4JRBB43KQCR","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":"b4d1db1a53075160464254cdde5f594fc1e6332ac478ac9504675b404d9a40ce","cross_cats_sorted":["cs.IT","math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-07-23T18:41:52Z","title_canon_sha256":"e66138c158f2248912843a72c7b0d56e5517ce55b72d3456d5dea3dde8482728"},"schema_version":"1.0","source":{"id":"1407.6329","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1407.6329","created_at":"2026-05-18T01:12:59Z"},{"alias_kind":"arxiv_version","alias_value":"1407.6329v1","created_at":"2026-05-18T01:12:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.6329","created_at":"2026-05-18T01:12:59Z"},{"alias_kind":"pith_short_12","alias_value":"UMAVNRWLPX5E","created_at":"2026-05-18T12:28:52Z"},{"alias_kind":"pith_short_16","alias_value":"UMAVNRWLPX5E3SE4","created_at":"2026-05-18T12:28:52Z"},{"alias_kind":"pith_short_8","alias_value":"UMAVNRWL","created_at":"2026-05-18T12:28:52Z"}],"graph_snapshots":[{"event_id":"sha256:6c7f0918a8eaf1c25a62cedc33dd9539a9097014537ce5bf3eb8ae1374911f8e","target":"graph","created_at":"2026-05-18T01:12:59Z","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":"We study $1$-perfect codes in Doob graphs $D(m,n)$. We show that such codes that are linear over $GR(4^2)$ exist if and only if $n=(4^{g+d}-1)/3$ and $m=(4^{g+2d}-4^{g+d})/6$ for some integers $g \\ge 0$ and $d>0$. We also prove necessary conditions on $(m,n)$ for $1$-perfect codes that are linear over $Z_4$ (we call such codes additive) to exist in $D(m,n)$ graphs; for some of these parameters, we show the existence of codes. For every $m$ and $n$ satisfying $2m+n=(4^t-1)/3$ and $m \\le (4^t-5\\cdot 2^{t-1}+1)/9$, we prove the existence of $1$-perfect codes in $D(m,n)$, without the restriction t","authors_text":"Denis Krotov (Sobolev Institute of Mathematics, Novosibirsk, Russia)","cross_cats":["cs.IT","math.IT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-07-23T18:41:52Z","title":"Perfect codes in Doob graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.6329","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":"sha256:08c3efa5820f580289eeaebf12edc5f2b6910ab9002551adc1866a1949d0d0b1","target":"record","created_at":"2026-05-18T01:12:59Z","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":"b4d1db1a53075160464254cdde5f594fc1e6332ac478ac9504675b404d9a40ce","cross_cats_sorted":["cs.IT","math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-07-23T18:41:52Z","title_canon_sha256":"e66138c158f2248912843a72c7b0d56e5517ce55b72d3456d5dea3dde8482728"},"schema_version":"1.0","source":{"id":"1407.6329","kind":"arxiv","version":1}},"canonical_sha256":"a30156c6cb7dfa4dc89c4c421e6d501443d89380f619adf2ef9aee2c0d07a30a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a30156c6cb7dfa4dc89c4c421e6d501443d89380f619adf2ef9aee2c0d07a30a","first_computed_at":"2026-05-18T01:12:59.821202Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:12:59.821202Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BYRAV9JHtG2h37OmMPpJ51C1PkijcFcUCSKQ/bZZa5PyFGJtGZ5THBTlVTvhwWGYRm+QxAt3mZGPZ3D0tD0nDg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:12:59.821550Z","signed_message":"canonical_sha256_bytes"},"source_id":"1407.6329","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:08c3efa5820f580289eeaebf12edc5f2b6910ab9002551adc1866a1949d0d0b1","sha256:6c7f0918a8eaf1c25a62cedc33dd9539a9097014537ce5bf3eb8ae1374911f8e"],"state_sha256":"a365e2d55c4703a09bf4488634603c7b0e0accb532ef5938387513f25dba0039"}