We derive the Gilbert-Varshamov and Hamming bounds for packings of spheres (codes) in the Grassmann manifolds over and. Asymptotic expressions are obtained for the geodesic metric and projection Frobenius (chordal) metric on the manifold.

PREFACE xv (for many interesting discussions of elliptic modules and modular curves); Gregory Katsman (who taught us coding theory); Leonid Bassalygo (who explained to us many coding subtleties); Gilles Lachaud (for many years of fruitful cooperation and hospitality, which helped us to write many chapters of the book); Alexander Barg (for many fruitful remarks and for the first version of tables of asymptotic bounds); Sergei Gelfand (for inducing us to publish our first paper improving the Gilbert-Varshamov bound); Gregory Kabatiansky (who made lots of valuable remarks reading the text of this book); Simon Litsyn (who attracted our attention to sphere packings in R n); Michael Rosenbloom (who worked with us on the problem of analogues); Alexei Skorobogatov (for many discussions); Andries Brouwer, Gerard van der Geer, and Marcel van der Vlught (who wrote appendices to the Russian edition of this book); members of the coding theory seminar and our colleagues from the Institute for Information Transmission Problems; and many other mathematicians for their friendship and help. Dmitry Nogin thanks not only the above-named mathematicians but also his co-authors, who attracted him to undertake this work. We are deeply grateful to our parents for their care and to our wives for their tender love.

We give new proofs of asymptotic upper bounds of coding theory obtained within the frame of Delsarte's linear programming method. The proofs rely on the analysis of eigenvectors of some finitedimensional operators related to orthogonal polynomials. The examples of the method considered in the paper include binary codes, binary constant-weight codes, spherical codes, and codes in the projective spaces. Introduction.Let X be a compact metric space with distance function d. A code C is a finite subset of X. Define the minimum distance of C as d(C) = min x,y∈C,x =y d(x, y). A variety of metric spaces that arise from different applications include the binary Hamming space, the binary Johnson space, the sphere in R n , real and complex projective spaces, Grassmann manifolds, etc. Estimating the maximum size of the code with a given value of d is one of the main problems of coding theory. Let M be the cardinality of C. A powerful technique to bound M above as a function of d(C) that is applicable in a wide class of metric spaces including all of the aforementioned examples is Delsarte's linear programming method [2]. The first such examples to be considered were the binary Hamming space H n = {0, 1} n and the Johnson space J n,w ⊂ H n which is formed by all the vectors of H n of Hamming weight w, with the distance given by the Hamming metric. The best currently known asymptotic estimates of the size of binary codes and binary constant weight codes were obtained in McEliece, Rodemich, Rumsey, Welch [11] and are called the MRRW bounds. Shortly thereafter, Kabatiansky and Levenshtein [7] established an analogous bound for codes on the unit sphere in R n with Euclidean metric and some related spaces. This paper also introduced a general approach to bounding the code size in distance-transitive metric spaces based on harmonic analysis of their isometry group. This approach was furthered in papers [8,10] which also explored the limits of Delsarte's method.In this paper we suggest a new proof method for linear programming upper bounds of coding theory. Our approach, which relies on the analysis of eigenvectors of some finite-dimensional operators related to orthogonal polynomials arguably makes some steps of the proofs conceptually more transparent then those previously known. We also consider some of the main examples mentioned above, The linear-algebraic ideas that we follow were introduced in a recent paper by Bachoc [1] in which a similar approach has been taken to establish an asymptotic bound for codes in the real Grassmann manifold. A bound on the code size.We assume that X is a distance-transitive space which means that its isometry group G acts doubly transitively on ordered pairs of points at a given distance. In this case the zonal spherical kernels K i (x, y) associated with irreducible regular representations of G depend only on the distance between x and y. In all the examples mentioned above, except for the Grassmann manifold, K i (x, y) can be expressed as a univariate polynomial p i (x) of degree i, where x =...

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