{"paper":{"title":"On the connection between correlation-immune functions and perfect 2-colorings of the Boolean n-cube","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Vladimir N. Potapov","submitted_at":"2011-01-19T07:03:25Z","abstract_excerpt":"A coloring of the Boolean $n$-cube is called perfect if, for every vertex $x$, the collection of the colors of the neighbors of $x$ depends only on the color of $x$. A Boolean function is called correlation-immune of degree $n-m$ if it takes the value 1 the same number of times for each $m$-face of the Boolean $n$-cube. In the present paper it is proven that each Boolean function $\\chi^S$ ($S\\subset E^n$) satisfies the inequality $${\\rm nei}(S)+ 2({\\rm cor}(S)+1)(1-\\rho(S))\\leq n,$$ where ${\\rm cor}(S)$ is the maximum degree of the correlation immunity of $\\chi^S$, ${\\rm nei} (S)= \\frac{1}{|S|"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.3627","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"}