  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.
