{"paper":{"title":"On the non-existence of linear perfect Lee codes: The Zhang-Ge condition and a new polynomial criterion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","math.CO","math.IT"],"primary_cat":"cs.IT","authors_text":"Claudio Qureshi","submitted_at":"2018-05-26T01:30:30Z","abstract_excerpt":"The Golomb-Welch conjecture (1968) states that there are no $e$-perfect Lee codes in $\\mathbb{Z}^n$ for $n\\geq 3$ and $e\\geq 2$. This conjecture remains open even for linear codes. A recent result of Zhang and Ge establishes the non-existence of linear $e$-perfect Lee codes in $\\mathbb{Z}^n$ for infinitely many dimensions $n$, for $e=3$ and $4$. In this paper we extend this result in two ways. First, using the non-existence criterion of Zhang and Ge together with a generalized version of Lucas' theorem we extend the above result for almost all $e$ (i.e. a subset of positive integers with densi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.10409","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"}