Nonexistence of tight spherical design of harmonic index 4

We give a new upper bound of the cardinality of a set of equiangular lines in $\R^n$ with a fixed angle $θ$ for each $(n,θ)$ 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 $n \geq 3$.

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