{"paper":{"title":"Linear codes over the ring $\\mathbb{Z}_4 + u\\mathbb{Z}_4 + v\\mathbb{Z}_4 + w\\mathbb{Z}_4 + uv\\mathbb{Z}_4 + uw\\mathbb{Z}_4 + vw\\mathbb{Z}_4 + uvw\\mathbb{Z}_4$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Aditya Purwa Santika, Bustomi, Djoko Suprijanto","submitted_at":"2019-04-25T01:31:12Z","abstract_excerpt":"We investigate linear codes over the ring $\\mathbb{Z}_4 + u\\mathbb{Z}_4 + v\\mathbb{Z}_4 + w\\mathbb{Z}_4 + uv\\mathbb{Z}_4 + uw\\mathbb{Z}_4 + vw\\mathbb{Z}_4 + uvw\\mathbb{Z}_4$, with conditions $u^2=u$, $v^2=v$, $w^2=w$, $uv=vu$, $uw=wu$ and $vw=wv.$ We first analyze the structure of the ring and then define linear codes over this ring. Lee weight and Gray map for these codes are defined and MacWilliams relations for complete, symmetrized, and Lee weight enumerators are obtained. The Singleton bound as well as maximum distance separable codes are also considered. Furthermore, cyclic and quasi-cyc"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.11117","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"}