Levenshtein, Vladimir I. Universal bounds for codes and designs. (English) Zbl 0983.94056 Pless, V. S. (ed.) et al., Handbook of coding theory. Vol. 1. Part 1: Algebraic coding. Vol. 2. Part 2: Connections, Part 3: Applications. Amsterdam: Elsevier. 499-648 (1998). From the text: [For the entire collection see Zbl 0907.94001.]This chapter of the handbook of coding theory is devoted to universal bounds for codes and designs. The contents include: Codes and designs in compact metric spaces (Parameters of codes in compact metric spaces, a system \(Q\) of orthogonal polynomials for a compact metric space, and a restricted \(\tau\)-design problem for systems of orthogonal polynomials); Polynomial metric spaces and extremum problems for the system \(Q\) of orthogonal polynomials (Inequalities for nonnegative definite functions, Polynomial metric spaces, and a \(\sigma\)-packing and \(\tau\)-design problem for systems of orthogonal polynomials); Duality in bounding optimal sizes of codes and designs in polynomial graphs (Polynomial graphs, Basic inequalities for code parameters based on annihilating polynomials, and Duality in bounding optimal sizes of codes and designs); Applications of orthogonal polynomials (Properties of orthogonal polynomials, Bounds on extreme roots of orthogonal polynomials, Properties of adjacent systems of orthogonal polynomials, Main theorem and consequences, and Applications to polynomial metric spaces and polynomial graphs); and Summary for the basic polynomial spaces (The unit Euclidean sphere and the projective spaces, The Hamming space, The Johnson space). Cited in 51 Documents MSC: 94B65 Bounds on codes 05B30 Other designs, configurations 51E05 General block designs in finite geometry 94-02 Research exposition (monographs, survey articles) pertaining to information and communication theory Keywords:parameters of codes in compact metric spaces; polynomial metric spaces; duality in bounding optimal sizes; bounds on extreme roots; packings; universal bounds; designs in compact metric spaces; orthogonal polynomials; extremum problems; polynomial graphs; Hamming space; Johnson space Citations:Zbl 0907.94001 PDFBibTeX XMLCite \textit{V. I. Levenshtein}, in: Handbook of coding theory. Vol. 1. Part 1: Algebraic coding. Vol. 2. Part 2: Connections, Part 3: Applications. Amsterdam: Elsevier. 499--648 (1998; Zbl 0983.94056)