Point-missing s-resolvable t-designs: infinite series of 4-designs with constant index

Abstract

The paper deals with t-designs that can be partitioned into s-designs, each missing a point of the underlying set, called point-missing s-resolvable t-designs, with emphasis on their applications in constructing t-designs. The problem considered may be viewed as a generalization of overlarge sets which are defined as a partition of all the left( {begin{array}{c}v +1 kend{array}}right) k-sets chosen from a (v+1)-set X into (v+1) mutually disjoint s-(v,k,delta ) designs, each missing a different point of X. Among others, it is shown that the existence of a point-missing (t-1)-resolvable t-(v,k,lambda ) design leads to the existence of a t-(v,k+1,lambda ') design. As a result, we derive various infinite series of 4-designs with constant index using overlarge sets of disjoint Steiner quadruple systems. These have parameters 4-(3^n,5,5), 4-(3^n+2,5,5) and 4-(2^n+1,5,5), for n ge 2, and were unknown until now. We also include a recursive construction of point-missing s-resolvable t-designs and its application.