{"paper":{"title":"Self-Dual Codes over $\\mathbb{Z}_2\\times (\\mathbb{Z}_2+u\\mathbb{Z}_2)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Hongwei Liu, Long Yu, Qiong Huang, Xiusheng Liu","submitted_at":"2016-10-04T09:05:01Z","abstract_excerpt":"In this paper, we study self-dual codes over $\\mathbb{Z}_2 \\times (\\mathbb{Z}_2+u\\mathbb{Z}_2) $, where $u^2=0$. Three types of self-dual codes are defined. For each type, the possible values $\\alpha,\\beta$ such that there exists a code $\\mathcal{C}\\subseteq \\mathbb{Z}_{2}^\\alpha\\times (\\mathbb{Z}_2+u\\mathbb{Z}_2)^\\beta$ are established. We also present several approaches to construct self-dual codes over $\\mathbb{Z}_2 \\times (\\mathbb{Z}_2+u\\mathbb{Z}_2) $. Moreover, the structure of two-weight self-dual codes is completely obtained for $\\alpha \\cdot\\beta\\neq 0$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.00900","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"}