Improvement to lotto design tables

Abstract

An (n, k, p, t) lotto design is a collection of k-subsets of a set X of n numbers wherein every p-subset of X must intersect at least one k-subset in t or more elements. L(n,k,p,t) is the minimum number of k-subsets which guarantees an intersection of at least t numbers between any p-subset of X and at least one of the k-subsets. To determine L(n,k,p,t) is the main goal of lotto design research. In previous work on lotto designs, other researchers used sequential algorithms to find bounds for L(n,k,p,t). We will determine the number of non-isomorphic optimal lotto designs on 5 or 6 blocks for n,k,p,t <= 20 and also improve lower bounds for L(n,k,p,t) >= 6 if possible by a more efficient implementation of a backtracking algorithm.