Tight [formula omitted]-designs on one shell of Johnson association schemes

Abstract

Each nontrivial shell (i.e., subconstituent) of the Johnson association scheme J(v,k) is known to be a commutative association scheme which is the product of two smaller Johnson association schemes. The concept of t-designs in one shell of J(v,k) was naturally defined and studied by Martin in the context of mixed t-designs. The purpose of this paper is to try to push this study a bit further. First we give a direct proof of the theorem that if (Y,w) is a relative t-design in J(v,k) on p shells then the part in each shell must be a weighted (t−p+1)-design in the shell. In particular, it is a mixed (t−p+1)-design in one shell if the weight function is constant on each shell. We also present another approach that makes use of the Terwilliger algebra, based on the work of Tanaka. This result is essentially proved by Martin in 1998, but the proof there contains a small gap which is repaired in this paper. We also study the existence problems of tight 2-, 3- and 4-designs on one shell of Johnson association scheme J(v,k) with small parameters, say v≤1000, thereby expanding the range of search in the original paper of Martin in 1998.