{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:2WNL2UFZTPU4CJUEVOV63MITA2","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":"6dc48b4fcb04c565ed16b1d5d8b4455c2f8d46cb53bb82e9bef3256eb8422fc9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-12-18T11:41:33Z","title_canon_sha256":"71bc230890abc4b334d14c45995fdd8f9919dda34bd8aa3f82f37b22b4dc44da"},"schema_version":"1.0","source":{"id":"1512.05908","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.05908","created_at":"2026-05-18T01:22:53Z"},{"alias_kind":"arxiv_version","alias_value":"1512.05908v2","created_at":"2026-05-18T01:22:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.05908","created_at":"2026-05-18T01:22:53Z"},{"alias_kind":"pith_short_12","alias_value":"2WNL2UFZTPU4","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_16","alias_value":"2WNL2UFZTPU4CJUE","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_8","alias_value":"2WNL2UFZ","created_at":"2026-05-18T12:29:02Z"}],"graph_snapshots":[{"event_id":"sha256:65ea715fff63e23bf5a3fc9a79b1a0ee14d481715980cc73f599de254dc3a185","target":"graph","created_at":"2026-05-18T01:22:53Z","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 describe odd-length-cube tilings of the n-dimensional q-ary torus what includes q-periodic integer lattice tilings of R^n. In the language of coding theory these tilings correspond to perfect codes with respect to the maximum metric. A complete characterization of the two-dimensional tillings is presented and in the linear case, a description of general matrices, isometry and isomorphism classes is provided. Several methods to construct perfect codes from codes of smaller dimension or via sections are derived. We introduce a special type of matrices (perfect matrices) which are in correspon","authors_text":"Claudio Qureshi, Sueli I. R. Costa","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-12-18T11:41:33Z","title":"On Cube Tilings of Tori and Classification of Perfect Codes in the Maximum Metric"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.05908","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":"sha256:2e98fa9ba78fc55b19a52c88d3f8dd3e52dbdfde883abd367ce15c06ae76e040","target":"record","created_at":"2026-05-18T01:22:53Z","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":"6dc48b4fcb04c565ed16b1d5d8b4455c2f8d46cb53bb82e9bef3256eb8422fc9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-12-18T11:41:33Z","title_canon_sha256":"71bc230890abc4b334d14c45995fdd8f9919dda34bd8aa3f82f37b22b4dc44da"},"schema_version":"1.0","source":{"id":"1512.05908","kind":"arxiv","version":2}},"canonical_sha256":"d59abd50b99be9c12684ababedb1130694e66e99e465bb1ad71f5123affd05d1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d59abd50b99be9c12684ababedb1130694e66e99e465bb1ad71f5123affd05d1","first_computed_at":"2026-05-18T01:22:53.674912Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:22:53.674912Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"aq9Fh0RbdUgN66fWl541NB03uBG4PsN3KuFTDw0PeaDZzzK9CxedxmpgMwY3jZN1fWBTZlKBuMZLvkIUwlhsDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:22:53.675461Z","signed_message":"canonical_sha256_bytes"},"source_id":"1512.05908","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2e98fa9ba78fc55b19a52c88d3f8dd3e52dbdfde883abd367ce15c06ae76e040","sha256:65ea715fff63e23bf5a3fc9a79b1a0ee14d481715980cc73f599de254dc3a185"],"state_sha256":"d3cbe1a2deb00d001ed7330d29c661d47bf15dedb28d18c88fd3c5a3eaa2f7d8"}