Asymptotic Bounds for General Covering Designs

Given five positive integers and t where and a t‐ general covering design is a pair where X is a set of n elements (called points) and a multiset of k‐subsets of X (called blocks) such that every p‐subset of X intersects at least λ blocks of in at least t points. In this article we continue the work carried out by Etzion, Wei, and Zhang [Des. Codes Cryptogr. 5 (1995), 217–239] on the asymptotic covering density of general covering designs. We will present combinatorial constructions leading to new upper bounds on the asymptotic covering density of 4‐(n, 4, 6, 1) general covering designs and 4‐ general covering designs with . The new bound on the asymptotic covering density of 4‐(n, 4, 6, 1) general covering designs is equivalent to a new lower bound for the Turán density .