Bounds on s-Distance Sets with Strength t

Abstract

A finite set $X$ in the Euclidean unit sphere is called an $s$-distance set if the set of distances between any two distinct elements of $X$ has size $s$. We say that $t$ is the strength of $X$ if $X$ is a spherical $t$-design but not a spherical $(t+1)$-design. Delsarte Goethals, and Seidel gave an absolute bound for the cardinality of an $s$-distance set. The results of Neumaier and Cameron, Goethals, and Seidel imply that if $X$ is a spherical $2$-distance set with strength $2$, then the known absolute bound for $2$-distance sets can be improved. This bound is also regarded as that for a strongly regular graph with a certain condition of the Krein parameters. In this paper, we give two generalizations of this bound to spherical $s$-distance sets with strength $t$ (more generally, to $s$-distance sets with strength $t$ in a two-point-homogeneous space) and to $Q$-polynomial association schemes. First, for any $s$ and $s-1 \leq t \leq 2s-2$, we improve the known absolute bound for the size of a spherical $s$-distance set with strength $t$. Second, for any $s$, we give an absolute bound for the size of a $Q$-polynomial association scheme of class $s$ with some conditions of the Krein parameters.