{"paper":{"title":"On the Most Informative Boolean Functions of the Very Noisy Channel","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Hengjie Yang, Richard D. Wesel","submitted_at":"2018-07-30T11:17:07Z","abstract_excerpt":"Let $X^n$ be a uniformly distributed $n$-dimensional binary vector, and $Y^n$ be the result of passing $X^n$ through a binary symmetric channel (BSC) with crossover probability $\\alpha$. A recent conjecture postulated by Courtade and Kumar states that for any Boolean function $f:\\{0,1\\}^n\\to\\{0,1\\}$, $I(f(X^n);Y^n)\\le 1-H(\\alpha)$. Although the conjecture has been proved to be true in the dimension-free high noise regime by Samorodnitsky, here we present a calculus-based approach to show a dimension-dependent result by examining the second derivative of $H(\\alpha)-H(f(X^n)|Y^n)$ at $\\alpha=1/2"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.11289","kind":"arxiv","version":3},"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"}