{"paper":{"title":"On the secrecy gain of $\\ell$-modular lattices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT","math.NT"],"primary_cat":"cs.IT","authors_text":"Anne-Maria Ernvall-Hyt\\\"onen, Esa V. Vesalainen","submitted_at":"2017-08-30T12:40:21Z","abstract_excerpt":"We show that for every $\\ell>1$, there is a counterexample to the $\\ell$-modular secrecy function conjecture by Oggier, Sol\\'e and Belfiore. These counterexamples all satisfy the modified conjecture by Ernvall-Hyt\\\"onen and Sethuraman. Furthermore, we provide a method to prove or disprove the modified conjecture for any given $\\ell$-modular lattice rationally equivalent to a suitable amount of copies of $\\mathbb{Z}\\oplus \\sqrt{\\ell}\\,\\mathbb{Z}$ with $\\ell \\in \\{3,5,7,11,23\\}$. We also provide a variant of the method for strongly $\\ell$-modular lattices when $\\ell\\in \\{6,14,15\\}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.09239","kind":"arxiv","version":3},"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"}