  We give a new upper bound of the cardinality of a set of equiangular lines in
$\R^n$ with a fixed angle $\theta$ for each $(n,\theta)$ satisfying certain
conditions. Our techniques are based on semi-definite programming methods for
spherical codes introduced by Bachoc--Vallentin [J.Amer.Math.Soc.2008]. As a
corollary to our bound, we show the nonexistence of spherical tight designs of
harmonic index 4 on $S^{n-1}$ with $n \geq 3$.
