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Let D = {B1, B2,…, Bb} be a finite family of k‐subsets (called blocks) of a v‐set X(v) = {1, 2,…, v} (with elements called points). Then D is a (v, k, t) covering design or covering if every t‐subset of X(v) is contained in at least one block of D. The number of blocks, b, is the size of the covering, and the minimum size of the covering is called the covering number, denoted C(v, k, t). This article is concerned with new constructions of coverings. The constructions improve many upper bounds on the covering number C(v, k, t) © 1998 John Wiley & Sons, Inc. J Combin Designs 6:21–41, 1998

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A collection C of k-subsets (called blocks) of a v v-set Xðv vÞ ¼ f1; 2; . . . ; v vg (with elements called points) is called a t-ðv v; k; m; Þ covering if for every m-subset M of Xðv vÞ there is a subcollection K of C with jKj ! such that every block K 2 K has at least t points in common with M. It is required that v v ! k ! t and v v ! m ! t. The minimum number of blocks in a t-ðv v; k; m; Þ covering is denoted by C ðv v; k; t; mÞ. We present some constructions producing the best known upper bounds on C ðv v; k; t; mÞ for k ¼ 6, a parameter of interest to lottery players.