On the number of mutually disjoint cyclic designs

Abstract

We denote by $LS[N](t,k,v)$ a large set of $t$-$(v,k,lambda)$ designs of size $N$‎, ‎which is a partition of all $k$-subsets of‎ ‎a $v$-set into $N$ disjoint $t$-$(v,k,lambda)$ designs and‎ ‎$N={v-t choose k-t}/lambda$‎. ‎We use the notation‎ ‎$N(t,v,k,lambda)$ as the maximum possible number of mutually‎ ‎disjoint cyclic $t$-$(v,k,lambda)$designs‎. ‎In this paper we give‎ ‎some new bounds for $N(2,29,4,3)$ and $N(2,31,4,2)$‎. ‎Consequently‎ ‎we present new large sets $LS[9](2,4,29)‎, ‎LS[13](2,4,29)$ and‎ ‎$LS[7](2,4,31)$‎, ‎where their existences were already known‎.