A new relative bound for equiangular lines and nonexistence of tight spherical designs of harmonic index 4

We give a new upper bound for the cardinality of a set of equiangular lines in Rn with a fixed common angle θ for each (n, θ) satisfying certain conditions. Our techniques are based on semidefinite programming methods for spherical codes introduced by Bachoc and Vallentin (2008). As a corollary to our bound, we show the nonexistence of spherical tight designs of harmonic index 4 on a sphere in Rn with n≥ 3.