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A (v, k, t) covering design, or covering, is a family of k-subsets, called 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). This paper gives three new methods for constructing good coverings: a greedy algorithm similar to Conway and Sloane's algorithm for lexicographic codes [6], and two methods that synthesize new coverings from preexisting ones. Using these new methods, together with results in the literature, we build tables of upper bounds on C(v, k, t) for v ≤ 32, k ≤ 16, and t ≤ 8.

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A (v, k, t) covering design, or covering, is a family of k-subsets, called 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 ( t )(1+o(1)) exist (as v Ä ). An earlier paper by the first three authors [4] gave new methods for constructing good coverings, 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 asymptotically optimal: For fixed k and t, the size of the coverings constructed matches Ro dl's bound. The paper also makes the o(1) error bound explicit, and gives some evidence for a much stronger bound.

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