{"paper":{"title":"Cyclic codes over $\\mathbb{Z}_4[u]/\\langle u^k\\rangle$ of odd length","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Cao Yuan, Li Qingguo","submitted_at":"2016-06-15T00:58:19Z","abstract_excerpt":"Let $R=\\mathbb{Z}_{4}[u]/\\langle u^k\\rangle=\\mathbb{Z}_{4}+u\\mathbb{Z}_{4}+\\ldots+u^{k-1}\\mathbb{Z}_{4}$ ($u^k=0$) where $k\\in \\mathbb{Z}^{+}$ satisfies $k\\geq 2$. For any odd positive integer $n$, it is known that cyclic codes over $R$ of length $n$ are identified with ideals of the ring $R[x]/\\langle x^{n}-1\\rangle$. In this paper, an explicit representation for each cyclic code over $R$ of length $n$ is provided and a formula to count the number of codewords in each code is given. Then a formula to calculate the number of cyclic codes over $R$ of length $n$ is obtained. Precisely, the dual "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.04601","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"}