On antipodal spherical t - designs of degree s with t ≥ 2 s − 3 Eiichi Bannai and Etsuko Bannai

Delsarte-Goethals-Seidel [9] studied finite subsets X in the unit sphere S = {x = (x1, x2, . . . , xn) ∈ R n | x1 + x 2 2 + · · · + x 2 n = 1} from the viewpoint of algebraic combinatorics. Here the following two parameters s = s(X) and t = t(X) play important roles. The s is called the degree of X and is defined as the number of distinct distances between two distinct elements of X, (then X is called an s-distance set), while t is called the strength of X and is usually defined as the largest t such that X becomes a t-design. Note that when we say X is a t-design, the strength of X might be larger than s(X). So our use of t is slightly ambiguous, but this convention is very useful and we believe that serious confusion will not occur. The important results due to Delsarte-Goethals-Seidel [9] are as follows (the leader is referred to [9, 1] for the definition of undefined terminologies): (i) We always have t ≤ 2s. (ii) If t ≥ s− 1, then X is distance invariant. (iii) If t ≥ 2s− 2, then X has the structure of Q-polynomial association scheme of class s. (iv) t = 2s if and only if X is a tight 2s-design. (v) t = 2s− 1 and X is antipodal, i.e., if x ∈ X then −x ∈ X, if and only if X is a tight (2s− 1)-design.