{"paper":{"title":"Trace reconstruction with $\\exp( O( n^{1/3} ) )$ samples","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.IT","math.ST","stat.TH"],"primary_cat":"math.PR","authors_text":"Fedor Nazarov, Yuval Peres","submitted_at":"2016-12-12T10:30:13Z","abstract_excerpt":"In the trace reconstruction problem, an unknown bit string $x \\in \\{0,1\\}^n$ is observed through the deletion channel, which deletes each bit of $x$ with some constant probability $q$, yielding a contracted string $\\widetilde{x}$. How many independent copies of $\\widetilde{x}$ are needed to reconstruct $x$ with high probability? Prior to this work, the best upper bound, due to Holenstein, Mitzenmacher, Panigrahy, and Wieder (2008), was $\\exp(\\widetilde{O}(n^{1/2}))$. We improve this bound to $\\exp(O(n^{1/3}))$ using statistics of individual bits in the output and show that this bound is sharp "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.03599","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"}