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The quantity g (k) (v) was introduced in [6] as the minimum number of blocks necessary in a pairwise balanced design on v elements, subject to the condition that the longest block has length k. Recently, we have needed to use all possibilities for such minimal covering designs, and we record all non-isomorphic solutions to the problem for v ≤ 13.

The minimum number of incomplete blocks required to cover, exactly λ times, all t-element subsets from a set V of cardinality v (v > t) is denoted by g(λ, t; v).

A directed triple system of order v, DTS(v), is a pair (V, B) where V is a set of v elements and B is a collection of ordered triples of distinct elements of V with the property that every ordered pair of distinct elements of V occurs in exactly one triple as a subsequence. A set of triples in a DTS(v) D is a defining set for D if it occurs in no other… (More)

This is a preprint of an article accepted for publication in Ars Combinatoria c 2003 (copyright owner as specified in the journal). Abstract The minimum number of incomplete blocks required to cover, exactly λ times, all t-element subsets from a set V of cardinality v (v > t) is denoted by g(λ, t; v).

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