P- and Q-Polynomial Association Schemes and Their Antipodal P-Polynomial Covers

We find a condition on the intersection numbers of any P- and Q -polynomial association scheme Y with diameter at least 3, that holds if Y has an antipodal P-polynomial cover with diameter at least 7. If Y is a known example of a P- and Q-polynomial association scheme with diameter at least 3, our condition holds iff Y has an antipodal P-polynomial cover with diameter at least 7.