{"paper":{"title":"Packing Sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Arne Winterhof, Ilya D. Shkredov, Oliver Roche-Newton","submitted_at":"2016-11-02T10:09:46Z","abstract_excerpt":"For a given subset $A\\subseteq \\mathbb F_q^*$, we study the problem of finding a large packing set $B$ of $A$, that is, a set $B \\subseteq \\mathbb F_q^*$ such that $|AB|=|A||B|$. We prove the existence of such a $B$ of size $|B|\\ge (q-1)/|A/A|$ and show that this bound is in general optimal.\n  The case that $q=p$ is a prime and $A=\\{1,2,\\ldots,\\lambda\\}$ for some positive integer $\\lambda$ is particularly interesting in view of the construction of limited-magnitude error correcting codes. Here we construct a packing set $B$ of size $|B|\\gg p (\\lambda \\log p)^{-1}$ for any $\\lambda \\le c p^{1/2"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.00529","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"}