Almost cover-free codes

We say that an $s$-subset of codewords of a code $X$ is ($(s,\ell)$-bad if $X$ contains $\ell$ other codewords such that the conjunction of these $\ell$ words is covered by the disjunction of the words of the $s$-subset. Otherwise, an $s$-subset of codewords of $X$ is said to be $(s,\ell)$-bad. A binary code $X$ is called a disjunctive $(s,\ell)$ cover-free (CF) code if $X$ does not contain $(s,\ell)$-bad subsets. We consider a probabilistic generalization of $(s,\ell)$ CF codes: we say that a binary code is an $(s,\ell)$ almost cover-free (ACF) code if almost all $s$-subsets of its codewords are $(s,\ell)$-good. The most interesting result is the proof of a lower and an upper bound for the capacity of $(s,\ell)$ ACF codes; the ratio of these bounds tends as $s\to\infty$ to the limit value $\log_2e/(\ell e)$.