×

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).

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

Citations:

Zbl 0907.94001
PDFBibTeX XMLCite