{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:JJNHMU2AP6YID4ZYEXHZIPEBUF","short_pith_number":"pith:JJNHMU2A","schema_version":"1.0","canonical_sha256":"4a5a7653407fb081f33825cf943c81a1593ec9c72ca4595f3d53962cba32092a","source":{"kind":"arxiv","id":"2505.08495","version":1},"attestation_state":"computed","paper":{"title":"On lattice tilings of $\\mathbb{Z}^n$ by limited magnitude error balls $\\mathcal{B}(n,2,k_{1},k_{2})$ with $k_1>k_2$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.IT","math.IT"],"primary_cat":"math.CO","authors_text":"Daohua Wang, Ka Hin Leung, Ran Tao, Tao Zhang","submitted_at":"2025-05-13T12:24:08Z","abstract_excerpt":"Lattice tilings of $\\mathbb{Z}^n$ by limited-magnitude error balls correspond to linear perfect codes under such error models and play a crucial role in flash memory applications. In this work, we establish three main results. First, we fully determine the existence of lattice tilings by $\\mathcal{B}(n,2,3,0)$ in all dimensions $n$. Second, we completely resolve the case $k_1=k_2+1$. Finally, we prove that for any integers $k_1>k_2\\ge0$ where $k_1+k_2+1$ is composite, no lattice tiling of $\\mathbb{Z}^n$ by the error ball $\\mathcal{B}(n,2,k_1,k_2)$ exists for sufficiently large $n$."},"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":"2505.08495","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-05-13T12:24:08Z","cross_cats_sorted":["cs.IT","math.IT"],"title_canon_sha256":"4af0788b627080d7f2355f0183c7b2cf29d286aa301375c22def9a45a67a4619","abstract_canon_sha256":"932b48f37cb0f26d4dad1eac42307c8b52fdc63eb19eefb28d3d4cb5c5f2f1e9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:02:31.227952Z","signature_b64":"O9ctatBEe7ofNz7VRBX8+3L4e2yQPO0v+tMYrIt5SP69Cg7pYGNaAJp9fqhEgolT6f10EpiMRpHRp6udtuMNAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4a5a7653407fb081f33825cf943c81a1593ec9c72ca4595f3d53962cba32092a","last_reissued_at":"2026-07-05T11:02:31.227440Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:02:31.227440Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On lattice tilings of $\\mathbb{Z}^n$ by limited magnitude error balls $\\mathcal{B}(n,2,k_{1},k_{2})$ with $k_1>k_2$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.IT","math.IT"],"primary_cat":"math.CO","authors_text":"Daohua Wang, Ka Hin Leung, Ran Tao, Tao Zhang","submitted_at":"2025-05-13T12:24:08Z","abstract_excerpt":"Lattice tilings of $\\mathbb{Z}^n$ by limited-magnitude error balls correspond to linear perfect codes under such error models and play a crucial role in flash memory applications. In this work, we establish three main results. First, we fully determine the existence of lattice tilings by $\\mathcal{B}(n,2,3,0)$ in all dimensions $n$. Second, we completely resolve the case $k_1=k_2+1$. Finally, we prove that for any integers $k_1>k_2\\ge0$ where $k_1+k_2+1$ is composite, no lattice tiling of $\\mathbb{Z}^n$ by the error ball $\\mathcal{B}(n,2,k_1,k_2)$ exists for sufficiently large $n$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.08495","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2505.08495/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2505.08495","created_at":"2026-07-05T11:02:31.227522+00:00"},{"alias_kind":"arxiv_version","alias_value":"2505.08495v1","created_at":"2026-07-05T11:02:31.227522+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2505.08495","created_at":"2026-07-05T11:02:31.227522+00:00"},{"alias_kind":"pith_short_12","alias_value":"JJNHMU2AP6YI","created_at":"2026-07-05T11:02:31.227522+00:00"},{"alias_kind":"pith_short_16","alias_value":"JJNHMU2AP6YID4ZY","created_at":"2026-07-05T11:02:31.227522+00:00"},{"alias_kind":"pith_short_8","alias_value":"JJNHMU2A","created_at":"2026-07-05T11:02:31.227522+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2605.09871","citing_title":"A proof of purely singular splitting conjecture","ref_index":18,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/JJNHMU2AP6YID4ZYEXHZIPEBUF","json":"https://pith.science/pith/JJNHMU2AP6YID4ZYEXHZIPEBUF.json","graph_json":"https://pith.science/api/pith-number/JJNHMU2AP6YID4ZYEXHZIPEBUF/graph.json","events_json":"https://pith.science/api/pith-number/JJNHMU2AP6YID4ZYEXHZIPEBUF/events.json","paper":"https://pith.science/paper/JJNHMU2A"},"agent_actions":{"view_html":"https://pith.science/pith/JJNHMU2AP6YID4ZYEXHZIPEBUF","download_json":"https://pith.science/pith/JJNHMU2AP6YID4ZYEXHZIPEBUF.json","view_paper":"https://pith.science/paper/JJNHMU2A","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2505.08495&json=true","fetch_graph":"https://pith.science/api/pith-number/JJNHMU2AP6YID4ZYEXHZIPEBUF/graph.json","fetch_events":"https://pith.science/api/pith-number/JJNHMU2AP6YID4ZYEXHZIPEBUF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JJNHMU2AP6YID4ZYEXHZIPEBUF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JJNHMU2AP6YID4ZYEXHZIPEBUF/action/storage_attestation","attest_author":"https://pith.science/pith/JJNHMU2AP6YID4ZYEXHZIPEBUF/action/author_attestation","sign_citation":"https://pith.science/pith/JJNHMU2AP6YID4ZYEXHZIPEBUF/action/citation_signature","submit_replication":"https://pith.science/pith/JJNHMU2AP6YID4ZYEXHZIPEBUF/action/replication_record"}},"created_at":"2026-07-05T11:02:31.227522+00:00","updated_at":"2026-07-05T11:02:31.227522+00:00"}