{"paper":{"title":"Lattices from codes over $\\mathbb{Z}_q$: Generalization of Constructions $D$, $D'$ and $\\overline{D}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Eleonesio Strey, Sueli I.R. Costa","submitted_at":"2015-12-18T01:40:00Z","abstract_excerpt":"In this paper, we extend the lattice Constructions $D$, $D'$ and $\\overline{D}$ $($this latter is also known as Forney's code formula$)$ from codes over $\\mathbb{F}_p$ to linear codes over $\\mathbb{Z}_q$, where $q \\in \\mathbb{N}$. We define an operation in $\\mathbb{Z}_q^n$ called zero-one addition, which coincides with the Schur product when restricted to $\\mathbb{Z}_2^n$ and show that the extended Construction $\\overline{D}$ produces a lattice if and only if the nested codes are closed under this addition. A generalization to the real case of the recently developed Construction $A'$ is also d"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.05841","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"}