C O ] 2 8 M ay 2 00 2 On Asymmetric Coverings and Covering Numbers

An asymmetric covering D(n,R) is a collection of special subsets S of an n-set such that every subset T of the n-set is contained in at least one special S with |S| − |T | ≤ R. In this paper we compute the smallest size of any D(n, 1) for n ≤ 8. We also investigate “continuous” and “banded” versions of the problem. The latter involves the classical covering numbers C(n, k, k−1), and we determine the following new values: C(10, 5, 4) = 51, C(11, 7, 6, ) = 84, C(12, 8, 7) = 126, C(13, 9, 8) = 185 and C(14, 10, 9) = 259. We also find the number of nonisomorphic minimal covering designs in several cases.