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We denote $R_k=F_{p^m}[u]/\\langle u^k\\rangle =F_{p^m}+uF_{p^m}+\\ldots+u^{k-1}F_{p^m}$ ($u^k=0$) and $\\lambda=a_0+a_1u+\\ldots+a_{k-1}u^{k-1}$ where $a_0, a_1,\\ldots, a_{k-1}\\in F_{p^m}$ satisfying $a_0\\neq 0$ and $a_1=1$. Let $r$ be a positive integer satisfying $p^{r-1}+1\\leq k\\leq p^r$. We defined a Gray map from $R_k$ to $F_{p^m}^{p^r}$ first, then prove that the Gray image of any linear $\\lambda$-constacyclic code over $R_k$ of length $N$ is a distance invariant linear $"},"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":"1610.01471","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-10-05T15:07:04Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"cdc46f3651baa1ea3d62b6171e83b9a5d32fb3d6ada85e9fb5b85b5911ee6711","abstract_canon_sha256":"6690b49e37068ef927020014a7d3b878fd090bf8b77a1c563290782378f52604"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:48:37.079449Z","signature_b64":"AbWyDR+8rUcqAsWEoI4aNN66K9awmksGddNQv+1Mql7Ic8DmHc/b3lFwt7yRkbB9l43LNJ7uOUz2m4ZgdtJdDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5866eb88ad0f6cb9ae8051b7b2695f4ff749c2de7ecb218b3c24c2c3505c68b5","last_reissued_at":"2026-05-18T00:48:37.078847Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:48:37.078847Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Gray image of constacyclic codes over the finite chain ring $F_{p^m}[u]/\\langle u^k\\rangle$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Yonglin Cao, Yuan Cao","submitted_at":"2016-10-05T15:07:04Z","abstract_excerpt":"Let $\\mathbb{F}_{p^m}$ be a finite field of cardinality $p^m$, where $p$ is a prime, and $k, N$ be any positive integers. We denote $R_k=F_{p^m}[u]/\\langle u^k\\rangle =F_{p^m}+uF_{p^m}+\\ldots+u^{k-1}F_{p^m}$ ($u^k=0$) and $\\lambda=a_0+a_1u+\\ldots+a_{k-1}u^{k-1}$ where $a_0, a_1,\\ldots, a_{k-1}\\in F_{p^m}$ satisfying $a_0\\neq 0$ and $a_1=1$. Let $r$ be a positive integer satisfying $p^{r-1}+1\\leq k\\leq p^r$. We defined a Gray map from $R_k$ to $F_{p^m}^{p^r}$ first, then prove that the Gray image of any linear $\\lambda$-constacyclic code over $R_k$ of length $N$ is a distance invariant linear $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.01471","kind":"arxiv","version":2},"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":"1610.01471","created_at":"2026-05-18T00:48:37.078933+00:00"},{"alias_kind":"arxiv_version","alias_value":"1610.01471v2","created_at":"2026-05-18T00:48:37.078933+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.01471","created_at":"2026-05-18T00:48:37.078933+00:00"},{"alias_kind":"pith_short_12","alias_value":"LBTOXCFNB5WL","created_at":"2026-05-18T12:30:29.479603+00:00"},{"alias_kind":"pith_short_16","alias_value":"LBTOXCFNB5WLTLUA","created_at":"2026-05-18T12:30:29.479603+00:00"},{"alias_kind":"pith_short_8","alias_value":"LBTOXCFN","created_at":"2026-05-18T12:30:29.479603+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/LBTOXCFNB5WLTLUAKG33E2K7J7","json":"https://pith.science/pith/LBTOXCFNB5WLTLUAKG33E2K7J7.json","graph_json":"https://pith.science/api/pith-number/LBTOXCFNB5WLTLUAKG33E2K7J7/graph.json","events_json":"https://pith.science/api/pith-number/LBTOXCFNB5WLTLUAKG33E2K7J7/events.json","paper":"https://pith.science/paper/LBTOXCFN"},"agent_actions":{"view_html":"https://pith.science/pith/LBTOXCFNB5WLTLUAKG33E2K7J7","download_json":"https://pith.science/pith/LBTOXCFNB5WLTLUAKG33E2K7J7.json","view_paper":"https://pith.science/paper/LBTOXCFN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1610.01471&json=true","fetch_graph":"https://pith.science/api/pith-number/LBTOXCFNB5WLTLUAKG33E2K7J7/graph.json","fetch_events":"https://pith.science/api/pith-number/LBTOXCFNB5WLTLUAKG33E2K7J7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LBTOXCFNB5WLTLUAKG33E2K7J7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LBTOXCFNB5WLTLUAKG33E2K7J7/action/storage_attestation","attest_author":"https://pith.science/pith/LBTOXCFNB5WLTLUAKG33E2K7J7/action/author_attestation","sign_citation":"https://pith.science/pith/LBTOXCFNB5WLTLUAKG33E2K7J7/action/citation_signature","submit_replication":"https://pith.science/pith/LBTOXCFNB5WLTLUAKG33E2K7J7/action/replication_record"}},"created_at":"2026-05-18T00:48:37.078933+00:00","updated_at":"2026-05-18T00:48:37.078933+00:00"}