ASYMPTOTICALLY OPTIMAL COVERING DESIGNS

A (v, k, t) covering design, or covering, is a family of k-subsets, ca lled blocks, chosen from a v-set, such that each t-subset is contained in at least one of the blocks. The number of blocks is the covering's size, and the minimum size of such a covering is denoted by C(v, k, t ). It is easy to see that a covering must contain at least ((v)(t))/(( k)(t)) blocks, and in 1985 Rodl [5] proved a long-standing conjecture of Erdos and Hanani [3] that for fixed k and t, coverings of size ((v) (t))/((k)(t))(1 + o(1)) exist (as v --> infinity). An earlier paper by the first three authors [4] gave new methods for constructing good co verings, and gave tables of upper bounds on C(v, k, t) for small v, k, and t. The present paper shows that two of those constructions are as ymptotically optimal: For fixed k and t, the size of the coverings con structed matches Rodl's bound. The paper also makes the oil) error bou nd explicit, and gives some evidence for a much stronger bound. (C) 19 96 Academic Press Inc.