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

Abstract

AbstractThe 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 $$\left( {\begin{array}{c}v +1\\ k\end{array}}\right) $$ v + 1 k k-sets chosen from a $$(v+1)$$ ( v + 1 ) -set X into $$(v+1)$$ ( v + 1 ) mutually disjoint s-$$(v,k,\delta )$$ ( v , k , δ ) designs, each missing a different point of X. Among others, it is shown that the existence of a point-missing $$(t-1)$$ ( t - 1 ) -resolvable t-$$(v,k,\lambda )$$ ( v , k , λ ) design leads to the existence of a t-$$(v,k+1,\lambda ')$$ ( v , k + 1 , λ ′ ) 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-$$(3^n,5,5)$$ ( 3 n , 5 , 5 ) , 4-$$(3^n+2,5,5)$$ ( 3 n + 2 , 5 , 5 ) and 4-$$(2^n+1,5,5)$$ ( 2 n + 1 , 5 , 5 ) , for $$n \ge 2$$ n ≥ 2 , and were unknown until now. We also include a recursive construction of point-missing s-resolvable t-designs and its application.