{"paper":{"title":"On Lattice Tilings of Asymmetric Limited-Magnitude Balls $\\cB(n,2,m,m-1)$","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hengjia Wei, Zhihao Guan, Ziqing Xiang","submitted_at":"2025-01-15T07:56:00Z","abstract_excerpt":"Limited-magnitude errors modify a transmitted integer vector in at most $t$ entries, where each entry can increase by at most $\\kp$ or decrease by at most $\\km$. This channel model is particularly relevant to applications such as flash memories and DNA storage. A perfect code for this channel is equivalent to a tiling of $\\Z^n$ by asymmetric limited-magnitude balls $\\cB(n,t,\\kp,\\km)$. In this paper, we focus on the case where $t=2$ and $\\km=\\kp-1$, and we derive necessary conditions on $m$ and $n$ for the existence of a lattice tiling of $\\cB(n,2,m,m-1)$. Specifically, we prove that if such a "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.08636","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2501.08636/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}