A functional view of upper bounds on codes

1. Introduction. In the problem of bounding the size of codes in compact homogeneous spaces, Del-sarte's polynomial method gives rise to the most powerful universal bounds on codes. Many overviews of the method exist in the literature; see for instance Levenshtein (1998). The purpose of this talk is to present a functional perspective of this method and give some examples. Let X be a compact metric space whose isometry group G acts transitively on it. The zonal polyno-mials associated with this action give rise to a family of orthogonal polynomials P(X) = {P κ } where κ = 0, 1,. .. is the total degree. These polynomials are univariate if G acts on X doubly transitively (the well-known examples include the Hamming and Johnson graphs and their q-analogs and other Q-polynomial distance-regular graphs; the sphere S n−1 ∈ R n) and are multivariate otherwise. First consider the univariate case. Then for any given value of the degree κ = i the family P(X) contains only one polynomial, denoted below by P i. Suppose that the distance on X is measured in such a way that d(x, x) = 1 and the diameter of X equals −1 (to accomplish this, a change of variable is made in the natural distance function on X). We refer to the model case of X = S n−1 although the arguments below apply to all spaces X with the above properties. Let f, g = 1 −1 f gdµ be the inner product in L 2 ([−1, 1], dµ) where dµ(x) is a distribution on [−1, 1] induced by an invariant measure on G. We assume that 1, 1 = 1. By Delsarte's fundamental theorem, the size of the code C ⊂ X whose distances take values in [−1, s] is bounded above by