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If $ p \\nmid \\ell$ then the ring $\\R:=\\O_K / p \\O_K$ is isomorphic to $\\F_{p^2}$ or $\\F_p \\times \\F_p$. Given a code $C$ over $\\R$, theta functions on the corresponding lattices are defined. These theta series $\\theta_{\\Lambda_{\\ell}(C)}$ can be written in terms of the complete weight enumerator of $C$. We show that for any two $\\ell < \\ell^\\prime$ the first $\\frac {"},"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":"1209.0475","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2012-09-03T20:07:19Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"05729df78b591bb00a981f664bcb0d17ce50134ea753f4d8fc07bd3bac0d063d","abstract_canon_sha256":"1eeecf60c52f7f53af8a3286b5a6259b6a0d16c009dad02459fbe0da4ddd2ff5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:46:18.035838Z","signature_b64":"8GqCgLI7e7QnQITAkJbmmVN/X2+YUkDmrgBC6V+G0Pk+nV7Sb2Gmd4ZsvJpQrKCrrrrX23PdasekFHSwoJLoAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9addbaf46c8c35fd17807f98feb4335b8cede7afc0022776c91a93bec7e9d566","last_reissued_at":"2026-05-18T03:46:18.035373Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:46:18.035373Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Codes over rings of size $p^2$ and lattices over imaginary quadratic fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.AG","authors_text":"C. Shor, G. Wijesiri, T. Shaska","submitted_at":"2012-09-03T20:07:19Z","abstract_excerpt":"Let $\\ell>0$ be a square-free integer congruent to 3 mod 4 and $\\O_K$ the ring of integers of the imaginary quadratic field $K=Q(\\sqrt{-\\ell})$. Codes $C$ over rings $\\O_K / p \\O_K$ determine lattices $\\Lambda_\\ell (C) $ over $K$. If $ p \\nmid \\ell$ then the ring $\\R:=\\O_K / p \\O_K$ is isomorphic to $\\F_{p^2}$ or $\\F_p \\times \\F_p$. Given a code $C$ over $\\R$, theta functions on the corresponding lattices are defined. These theta series $\\theta_{\\Lambda_{\\ell}(C)}$ can be written in terms of the complete weight enumerator of $C$. We show that for any two $\\ell < \\ell^\\prime$ the first $\\frac {"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.0475","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":"1209.0475","created_at":"2026-05-18T03:46:18.035446+00:00"},{"alias_kind":"arxiv_version","alias_value":"1209.0475v1","created_at":"2026-05-18T03:46:18.035446+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.0475","created_at":"2026-05-18T03:46:18.035446+00:00"},{"alias_kind":"pith_short_12","alias_value":"TLO3V5DMRQ27","created_at":"2026-05-18T12:27:23.164592+00:00"},{"alias_kind":"pith_short_16","alias_value":"TLO3V5DMRQ272F4A","created_at":"2026-05-18T12:27:23.164592+00:00"},{"alias_kind":"pith_short_8","alias_value":"TLO3V5DM","created_at":"2026-05-18T12:27:23.164592+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/TLO3V5DMRQ272F4AP6MP5NBTLO","json":"https://pith.science/pith/TLO3V5DMRQ272F4AP6MP5NBTLO.json","graph_json":"https://pith.science/api/pith-number/TLO3V5DMRQ272F4AP6MP5NBTLO/graph.json","events_json":"https://pith.science/api/pith-number/TLO3V5DMRQ272F4AP6MP5NBTLO/events.json","paper":"https://pith.science/paper/TLO3V5DM"},"agent_actions":{"view_html":"https://pith.science/pith/TLO3V5DMRQ272F4AP6MP5NBTLO","download_json":"https://pith.science/pith/TLO3V5DMRQ272F4AP6MP5NBTLO.json","view_paper":"https://pith.science/paper/TLO3V5DM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1209.0475&json=true","fetch_graph":"https://pith.science/api/pith-number/TLO3V5DMRQ272F4AP6MP5NBTLO/graph.json","fetch_events":"https://pith.science/api/pith-number/TLO3V5DMRQ272F4AP6MP5NBTLO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TLO3V5DMRQ272F4AP6MP5NBTLO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TLO3V5DMRQ272F4AP6MP5NBTLO/action/storage_attestation","attest_author":"https://pith.science/pith/TLO3V5DMRQ272F4AP6MP5NBTLO/action/author_attestation","sign_citation":"https://pith.science/pith/TLO3V5DMRQ272F4AP6MP5NBTLO/action/citation_signature","submit_replication":"https://pith.science/pith/TLO3V5DMRQ272F4AP6MP5NBTLO/action/replication_record"}},"created_at":"2026-05-18T03:46:18.035446+00:00","updated_at":"2026-05-18T03:46:18.035446+00:00"}