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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...

In this paper, we introduce and solve some new problems of efficient reconstruction of an unknown sequence from its versions distorted by errors of a certain type. These erroneous versions are considered as outputs of repeated transmissions over a channel, either combinatorial channel defined by the maximum number of permissible errors of a given type, or a discrete memoryless channel. We are interested in the smallest such that erroneous versions always suffice to reconstruct a sequence of length , either exactly or with a preset accuracy and/or with a given probability. We are also interested in simple reconstruction algorithms. Complete solutions for combinatorial channels with some types of errors of interest in coding theory, namely, substitutions, transpositions, deletions, and insertions of symbols are given. For these cases, simple reconstruction algorithms based on majority and threshold principles and their nontrivial combination are found. In general, for combinatorial channels the considered problem is reduced to a new problem of reconstructing a vertex of an arbitrary graph with the help of the minimum number of vertices in its metrical ball of a given radius. A certain sufficient condition for solution of this problem is presented. For a discrete memoryless channel, asymptotic behavior of the minimum number of repeated transmissions which are sufficient to reconstruct any sequence of length within Hamming distance with error probability is found when and tend to 0 as. A similar result for the continuous channel with discrete time and additive Gaussian noise is also obtained.

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