Tight spherical 5-designs of minimal type exist exactly when a specific Q-polynomial coherent configuration exists, and they cannot exist in infinitely many dimensions including 119 and 527.
Munemasa, Spherical Designs, in: Handbook of Combinatorial Designs, 2nd ed., CRC Press, pp
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Existence and nonexistence of spherical $5$-designs of minimal type
Tight spherical 5-designs of minimal type exist exactly when a specific Q-polynomial coherent configuration exists, and they cannot exist in infinitely many dimensions including 119 and 527.