Minimum Coverings Of Triples By Quadruples

A t-(v, k,λ) covering design is a pair (X,β), where X is a v-set of element (points) andβis a collection of k-subsets (blocks) of X, such that each t-subset of X occurs in at leastλblocks inβ, where t is called strength. The t-(v, k,λ) covering design is very important in design theory. The covering number C_λ(v, k, t) is the number of blocks in a minimum t-(v, k, A) covering design. For t = 2, much work has been done on covering numbers C_λ(v, k, 2) (see [1]). However, for t > 2, not much is known.In this paper, we use GDD designs, s-fan designs and Hartman's fundamental construction for 3-designs to obtain some CQSs or candelabra quadruple covering designs. They, together with holey quadruple covering designs, are used to determine the covering numbers, i.e.,with exceptions of C₂(7,4,3) = [7/4C₂(6,3,2)] + 2 and possible exceptions ofλ≡1 (mod 4) and v = 12k + 7, k∈{1,2,3,4, 5, 7,8,9,10,11,12,16, 21,23,25, 29}; v = 27 andλ= 2,5,7 (mod 12); v = 35 andλ= 3 (mod 4) orλ= 2.