Uniqueness of certain association schemes

We prove the uniqueness of the two association schemes which appear in recent work of Henry Cohn and others in connection with their study of universally optimal spherical codes in Euclidean spaces: one is the class 4 association scheme with 40 vertices in R 10 and the other one is the class 3 association scheme with 64 vertices in R 14 . We prove the uniqueness mainly by geometric considerations with some computational help.