{"paper":{"title":"Covering sets for limited-magnitude errors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT","math.NT"],"primary_cat":"cs.IT","authors_text":"Arne Winterhof, Igor E. Shparlinski, Zhixiong Chen","submitted_at":"2013-10-01T02:14:04Z","abstract_excerpt":"For a set\n  $\\cM=\\{-\\mu,-\\mu+1,\\ldots, \\lambda\\}\\setminus\\{0\\}$ with non-negative integers $\\lambda,\\mu<q$ not both 0, a subset $\\cS$ of the residue class ring $\\Z_q$ modulo an integer $q\\ge 1$ is called a $(\\lambda,\\mu;q)$-\\emph{covering set} if $$ \\cM \\cS=\\{ms \\bmod q : m\\in \\cM,\\ s\\in \\cS\\}=\\Z_q. $$ Small covering sets play an important role in codes correcting limited-magnitude errors. We give an explicit construction of a $(\\lambda,\\mu;q)$-covering set $\\cS$ which is of the size $q^{1 + o(1)}\\max\\{\\lambda,\\mu\\}^{-1/2}$ for almost all integers $q\\ge 1$ and of optimal size $p\\max\\{\\lambda,\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.0120","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"}