{"paper":{"title":"Almost Cover-Free Codes and Designs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Arkadii D'yachkov, Ilya Vorobyev, Nikita Polyanskii, Vladislav Shchukin","submitted_at":"2014-10-30T21:30:42Z","abstract_excerpt":"An $s$-subset of codewords of a binary code $X$ is said to be an {\\em $(s,\\ell)$-bad} in $X$ if the code $X$ contains a subset of other $\\ell$ codewords such that the conjunction of the $\\ell$ codewords is covered by the disjunctive sum of the $s$ codewords. Otherwise, the $s$-subset of codewords of $X$ is said to be an {\\em $(s,\\ell)$-good} in~$X$.mA binary code $X$ is said to be a cover-free $(s,\\ell)$-code if the code $X$ does not contain $(s,\\ell)$-bad subsets. In this paper, we introduce a natural {\\em probabilistic} generalization of cover-free $(s,\\ell)$-codes, namely: a binary code is "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.8566","kind":"arxiv","version":5},"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"}