{"paper":{"title":"Codes over rings of size $p^2$ and lattices over imaginary quadratic fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.AG","authors_text":"C. Shor, G. Wijesiri, T. Shaska","submitted_at":"2012-09-03T20:07:19Z","abstract_excerpt":"Let $\\ell>0$ be a square-free integer congruent to 3 mod 4 and $\\O_K$ the ring of integers of the imaginary quadratic field $K=Q(\\sqrt{-\\ell})$. Codes $C$ over rings $\\O_K / p \\O_K$ determine lattices $\\Lambda_\\ell (C) $ over $K$. If $ p \\nmid \\ell$ then the ring $\\R:=\\O_K / p \\O_K$ is isomorphic to $\\F_{p^2}$ or $\\F_p \\times \\F_p$. Given a code $C$ over $\\R$, theta functions on the corresponding lattices are defined. These theta series $\\theta_{\\Lambda_{\\ell}(C)}$ can be written in terms of the complete weight enumerator of $C$. We show that for any two $\\ell < \\ell^\\prime$ the first $\\frac {"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.0475","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"}