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 distinct two 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-Seidel gave an absolute bound for the cardinality of an $s$-distance set. The results of Neumaier and Cameron-Goethals-Seidel imply that if $X$ is a spherical 2-distance set with strength 2, then the known absolute bound for 2-distance sets is improved. This bound are also regarded as that for a strongly regular graph with the 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 $t$. Secondly, for any $d$, we give an absolute bound for the size of a $Q$-polynomial association scheme of class $d$ with the certain conditions of the Krein parameters.