Bounds on the rate of disjunctive codes

A binary code is said to be a disjunctive $(s,\ell)$ cover-free code if it is an incidence matrix of a family of sets where the intersection of any $\ell$ sets is not covered by the union of any other $s$ sets of this family. A binary code is said to be a list-decoding disjunctive of strength $s$ with list size $L$ if it is an incidence matrix of a family of sets where the union of any $s$ sets can cover no more that $L-1$ other sets of this family. For $L=\ell=1$, both definitions coincide, and the corresponding binary code is called a disjunctive $s$-code. This paper is aimed at improving previously known and obtaining new bounds on the rate of these codes. The most interesting of the new results is a lower bound on the rate of disjunctive $(s,\ell)$ cover-free codes obtained by random coding over the ensemble of binary constant-weight codes; its ratio to the best known upper bound converges as $s\to\infty$, with an arbitrary fixed $\ell\ge1$, to the limit $2e^{-2}=0{,}271\dots$ In the classical case of $\ell=1$, this means that the upper bound on the rate of disjunctive $s$-codes constructed in 1982 by D'yachkov and Rykov is asymptotically attained up to a constant factor $a$, $2e^{-2}\le a\le1$.