ec 2 00 5 SPECTRAL APPROACH TO LINEAR PROGRAMMING BOUNDS ON CODES

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 finite-dimensional 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. 1. 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 [10] and are called the MRRW bounds. Shortly thereafter, Kabatiansky and Levenshtein [6] 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 [7, 9] 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 …