Equivalence of Delsarte's bounds for codes and designs in symmetric association schemes, and some applications

Abstract

In order to obtain bounds on the sizes of codes and designs in association schemes Delsarte introduced two extremum problems for the systems of p-numbers and q-numbers. We prove that the Delsarte bound for codes obtained with the help of either of these systems is equivalent to that for designs obtained by using the other system. In particular, this means that the universal Delsarte's bound of 1973 for designs is equivalent to the sphere packing bound for codes. Furthermore, the universal bound for codes obtained by the author in 1978 gives rise to a new universal bound for designs, in particular, for block designs. This bound improves upon the known bounds when the strength of the design is sufficiently large. Moreover, a relationship between bounds for orthogonal arrays and block designs is obtained which gives new lower bounds on the size of orthogonal arrays with the help of those for block designs.