On antipodal spherical t-designs of degree s with $t\geq 2s-3$
  We prove that if X is a spherical t-design and s-distance set with $t\geq 2s-3$, then X has the structure of Q-polynomial association scheme of class s. Also, we describe the parameters of the association scheme. 
Eiichi Bannai and Etsuko Bannai