{"paper":{"title":"Left dihedral codes over Galois rings ${\\rm GR}(p^2,m)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT","math.RA"],"primary_cat":"cs.IT","authors_text":"Fang-Wei Fu, Yonglin Cao, Yuan Cao","submitted_at":"2016-09-14T00:54:06Z","abstract_excerpt":"Let $D_{2n}=\\langle x,y\\mid x^n=1, y^2=1, yxy=x^{-1}\\rangle$ be a dihedral group, and $R={\\rm GR}(p^2,m)$ be a Galois ring of characteristic $p^2$ and cardinality $p^{2m}$ where $p$ is a prime. Left ideals of the group ring $R[D_{2n}]$ are called left dihedral codes over $R$ of length $2n$, and abbreviated as left $D_{2n}$-codes over $R$. Let ${\\rm gcd}(n,p)=1$ in this paper. Then any left $D_{2n}$-code over $R$ is uniquely decomposed into a direct sum of concatenated codes with inner codes ${\\cal A}_i$ and outer codes $C_i$, where ${\\cal A}_i$ is a cyclic code over $R$ of length $n$ and $C_i$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.04083","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"}