{"paper":{"title":"On the tightness of Tiet\\\"av\\\"ainen's bound for distributions with limited independence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Louay Bazzi","submitted_at":"2017-07-03T13:55:27Z","abstract_excerpt":"In 1990, Tiet\\\"av\\\"ainen showed that if the only information we know about a linear code is its dual distance $d$, then its covering radius $R$ is at most $\\frac{n}{2}-(\\frac{1}{2}-o(1))\\sqrt{dn}$. While Tiet\\\"av\\\"ainen's bound was later improved for large values of $d$, it is still the best known upper bound for small values including the $d = o(n)$ regime. Tiet\\\"av\\\"ainen's bound holds also for $(d-1)$-wise independent probability distributions on $\\{0,1\\}^n$, of which linear codes with dual distance $d$ are special cases. We show that Tiet\\\"av\\\"ainen's bound on $R-\\frac{n}{2}$ is asymptotic"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.00552","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"}