New lower bounds for $t$-coverings

Abstract

Fisher proved in 1940 that any $2$-$(v,k,\lambda )$ design with $v>k$ has at least $v$ blocks. In 1975 Ray-Chaudhuri and Wilson generalised this result by showing that every $t$-$(v,k,\lambda )$ design with $v\ge k+\lfloor t/2\rfloor$ has at least $\left(\genfrac{}{}{0ex}{}{v}{\lfloor t/2\rfloor }\right)$ blocks. By combining methods used by Bose and Wilson in proofs of these results, we obtain new lower bounds on the size of $t$-$(v,k,\lambda )$ coverings. Our results generalise lower bounds on the size of $2$-$(v,k,\lambda )$ coverings recently obtained by the first author.

19 pages, 0 figures