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If $|V|=2^d$, then $V$ is called a cube tiling code. Cube tiling codes determine $2$-periodic cube tilings of $\\mathbb{R}^d$ or, equivalently, tilings of the flat torus $\\mathbb{T}^d=\\{(x_1,\\ldots ,x_d)({\\rm mod} 2):(x_1,\\ldots ,x_d)\\in \\mathbb{R}^d\\}$ by translates of the unit cube as well as $r$-perf"},"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":"1805.07806","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-20T17:53:30Z","cross_cats_sorted":[],"title_canon_sha256":"fb4cd30f3fd44fa1c51f8802c503bc95a4f4d84035a84fad84c1bd4725265368","abstract_canon_sha256":"726aae31ea13a8cb9139d06ab14dab8f09c2d5983a6d824b8507a07a19955b5f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:15:32.491049Z","signature_b64":"RNcUsd97qzzKOby05OgJEJu96MjpP+Ikl48lRUFUBPPSeAJcmLBQ2Lv+tnt1se/VRrmweLnZdovSmkHpmNtBAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7096306638f62ac9b2034507d6914e1dd5f3663f024ea468fbe8202787f4fa66","last_reissued_at":"2026-05-18T00:15:32.490494Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:15:32.490494Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the structure of cube tiling codes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Andrzej P. Kisielewicz","submitted_at":"2018-05-20T17:53:30Z","abstract_excerpt":"Let $S$ be a set of arbitrary objects, and let $S^d=\\{v_1...v_d\\colon v_i\\in S\\}$. A polybox code is a set $V\\subset S^d$ with the property that for every two words $v,w\\in V$ there is $i\\in [d]$ with $v_i'=w_i$, where a permutation $s\\mapsto s'$ of $S$ is such that $s''=(s')'=s$ and $s'\\neq s$. If $|V|=2^d$, then $V$ is called a cube tiling code. Cube tiling codes determine $2$-periodic cube tilings of $\\mathbb{R}^d$ or, equivalently, tilings of the flat torus $\\mathbb{T}^d=\\{(x_1,\\ldots ,x_d)({\\rm mod} 2):(x_1,\\ldots ,x_d)\\in \\mathbb{R}^d\\}$ by translates of the unit cube as well as $r$-perf"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.07806","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":"1805.07806","created_at":"2026-05-18T00:15:32.490594+00:00"},{"alias_kind":"arxiv_version","alias_value":"1805.07806v1","created_at":"2026-05-18T00:15:32.490594+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.07806","created_at":"2026-05-18T00:15:32.490594+00:00"},{"alias_kind":"pith_short_12","alias_value":"OCLDAZRY6YVM","created_at":"2026-05-18T12:32:43.782077+00:00"},{"alias_kind":"pith_short_16","alias_value":"OCLDAZRY6YVMTMQD","created_at":"2026-05-18T12:32:43.782077+00:00"},{"alias_kind":"pith_short_8","alias_value":"OCLDAZRY","created_at":"2026-05-18T12:32:43.782077+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/OCLDAZRY6YVMTMQDIUD5NEKODX","json":"https://pith.science/pith/OCLDAZRY6YVMTMQDIUD5NEKODX.json","graph_json":"https://pith.science/api/pith-number/OCLDAZRY6YVMTMQDIUD5NEKODX/graph.json","events_json":"https://pith.science/api/pith-number/OCLDAZRY6YVMTMQDIUD5NEKODX/events.json","paper":"https://pith.science/paper/OCLDAZRY"},"agent_actions":{"view_html":"https://pith.science/pith/OCLDAZRY6YVMTMQDIUD5NEKODX","download_json":"https://pith.science/pith/OCLDAZRY6YVMTMQDIUD5NEKODX.json","view_paper":"https://pith.science/paper/OCLDAZRY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1805.07806&json=true","fetch_graph":"https://pith.science/api/pith-number/OCLDAZRY6YVMTMQDIUD5NEKODX/graph.json","fetch_events":"https://pith.science/api/pith-number/OCLDAZRY6YVMTMQDIUD5NEKODX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OCLDAZRY6YVMTMQDIUD5NEKODX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OCLDAZRY6YVMTMQDIUD5NEKODX/action/storage_attestation","attest_author":"https://pith.science/pith/OCLDAZRY6YVMTMQDIUD5NEKODX/action/author_attestation","sign_citation":"https://pith.science/pith/OCLDAZRY6YVMTMQDIUD5NEKODX/action/citation_signature","submit_replication":"https://pith.science/pith/OCLDAZRY6YVMTMQDIUD5NEKODX/action/replication_record"}},"created_at":"2026-05-18T00:15:32.490594+00:00","updated_at":"2026-05-18T00:15:32.490594+00:00"}