Designs as maximum codes in polynomial metric spaces

Abstract

Finite and infinite metric spaces ffJt that are polynomial with respect to a monotone substitution of variable t(d) are considered. A finite subset (code) W C_ ~JI is characterized by the minimal distance d(W) between its distinct elements, by the number l(W) of distances between its distinct elements and by the maximal strength r(W) of the design generated by the code W. A code W _C ~Jt is called a maximum one if it has the greatest cardinality among subsets of with minimal distance at least d(W), and diametrical if the diameter of W is equal to the diameter of the whole space 9~. Delsarte codes are codes 1V C ~ with v(W) ~ 21(W) -1 or r(W) = 2I(W) -2 > 0 and W is a diametrical code. It is shown that all parameters of Delsarte codes W C fl/t are uniquely determined by their cardinality IWI or minimal distance d(W) and that the minimal polynomials of the Delsarte codes W C ~ are expansible with positive coefficients in an orthogonal system of polynomials {Qi(t)} corresponding to 93L The main results of the present paper consist in a proof of maximality of all Delsarte codes provided that the system {Qi(t)} satisfies some condition and of a new proof confirming in this case the validity of all the results on the upper bounds for the maximum cardinallty of codes W C ~ with a given minimal distance, announced by the author in 1978. Moreover, it appeared that this condition is satisfied for all infinite polynomial metric spaces as well as for distance-regular graphs, decomposable in a sense defined below. It is also proved that with one exception all classical distance-regular graphs are decomposable. In addition for decomposable distance-regular graphs an improvement of the absolute Delsarte bound for diametrical codes is obtained. For the Hamming and Johnson spaces, Euclidean sphere, real and complex projective spaces, tables containing parameters of known Delsarte codes are presented. Moreover, for each of the above-mentioned infinite spaces infinite sequences (of maximum) Delsarte codes not belonging to tight designs are indicated. Known Bounds and New Results Concerning Polynomial Metric SpacesBy polynomial metric spaces we mean finite metric spaces represented by P-and Q-polynomial association schemes (Delsarte [8]) as well as infinite metric spaces that are connected compact two-point (or strongly) homogeneous ones [58, 29,49]. Thus finite polynomial metric spaces are actually Q-polynomial distance-regular graphs [1,4], simply called polynomial below. The best known examples of finite polynomial metric spaces are given by Hamming, Johnson, and Grassmann spaces. The Euclidean sphere, real and complex projective spaces present examples of infinite polynomial metric spaces. Every polynomial metric space ~rt is characterized by its metric d(x, y), diameter D = D(9)I) = max~.~cg~ ~ d(x, y), and a normalized measure/z~ according to which the measure of closed metric balls of arbitrary radius d (0 ~< d ~< D) does not depend on their centers a~d thus may be written as p(d) (#(D) = 1). In case of polynomial distance-re...

Universally optimal distribution of points on spheres

“…By contrast, E 8 yields a 3-design in RP 7 . The following theorem generalizes Theorem 1.2 and the results of Levenshtein from [Lev2] (namely, that all sharp configurations are optimal codes): Theorem 8.2. Let f : [−1, 1) → R be absolutely monotonic, and let C ⊂ X be a sharp configuration.…”

“…Recall that a metric space is two-point homogeneous if for each r > 0, its isometry group acts transitively on ordered pairs of points at distance r. In this section we generalize Theorem 1.2 to all compact, connected two-point homogeneous spaces. This generalization is parallel to the general setting for linear programming bounds in [CS,Lev2,Lev3]. It seems plausible that our results also generalize to the case of discrete two-point homogeneous spaces (see Chapter 9 of [CS] for examples), but we have not investigated that possibility.…”

Weighted complex projective 2-designs from bases: Optimal state determination by orthogonal measurements

“…For a unified treatment of designs in terms of metric spaces consult the work of Levenshtein [48,49,50] (see also Ref. 's [51,52,53,54,55,56,57,58]).…”

“…Theorem 2.2 is in fact a special case from known results within the theory of t-designs. In general, the number of design points must satisfy [42,46,48,51] 5) with equality only if the design has uniform weight [48], i.e. w(x) = 1/|D|.…”

Uniqueness of the 120-point spherical 11-design in four dimensions

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