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For any $\\delta,\\alpha\\in \\mathbb{F}_{q}^{\\times}$, an explicit representation for all distinct $(\\delta+\\alpha u^2)$-constacyclic codes over $R$ of length $n$ is given, and the dual code for each of these codes is determined. For the case of $q=2^m$ and $\\delta=1$, all self-dual $(1+\\alpha u^2)$-constacyclic codes over $R$ of odd"},"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":"1511.02369","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2015-11-07T16:07:42Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"1363796a32b03cda5ed261818146e238820ee2f4c40f09f2cc968ba47f9679c2","abstract_canon_sha256":"cb7be55f486375feb50d028cb2f15bb0427d5916c4534e8da83d8aebb852f7dd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:27:30.493531Z","signature_b64":"eU5jGrvUMM5CRApNZabypo5lBJZp+40yauBwWkJraJU+8XUbAq60CbjOinYDVcrj609866vZ9/EZ72ovyJL7Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0da68a4d8507e96511b39e3e2efec934e67ecbcf940d51b727c342cc25faf83e","last_reissued_at":"2026-05-18T01:27:30.492975Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:27:30.492975Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On a class of $(\\delta+\\alpha u^2)$-constacyclic codes over $\\mathbb{F}_{q}[u]/\\langle u^4\\rangle$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Jian Gao, Yonglin Cao, Yuan Cao","submitted_at":"2015-11-07T16:07:42Z","abstract_excerpt":"Let $\\mathbb{F}_{q}$ be a finite field of cardinality $q$, $R=\\mathbb{F}_{q}[u]/\\langle u^4\\rangle=\\mathbb{F}_{q}+u\\mathbb{F}_{q}+u^2\\mathbb{F}_{q}+u^3\\mathbb{F}_{q}$ $(u^4=0)$ which is a finite chain ring, and $n$ be a positive integer satisfying ${\\rm gcd}(q,n)=1$. For any $\\delta,\\alpha\\in \\mathbb{F}_{q}^{\\times}$, an explicit representation for all distinct $(\\delta+\\alpha u^2)$-constacyclic codes over $R$ of length $n$ is given, and the dual code for each of these codes is determined. 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