Coding Theory and Design Theory: Part I Coding Theory

ISBN-13: 9781461389965 / Angielski / Miękka / 2011 / 239 str.

Dijen Ray-Chaudhuri

Opis

This IMA Volume in Mathematics and its Applications Coding Theory and Design Theory Part I: Coding Theory is based on the proceedings of a workshop which was an integral part of the 1987-88 IMA program on APPLIED COMBINATORICS. We are grateful to the Scientific Committee: Victor Klee (Chairman), Daniel Kleitman, Dijen Ray-Chaudhuri and Dennis Stanton for planning and implementing an exciting and stimulating year long program. We especially thank the Workshop Organizer, Dijen Ray-Chaudhuri, for organizing a workshop which brought together many of the major figures in a variety of research fields in which coding theory and design theory are used. A vner Friedman Willard Miller, Jr. PREFACE Coding Theory and Design Theory are areas of Combinatorics which found rich applications of algebraic structures. Combinatorial designs are generalizations of finite geometries. Probably, the history of Design Theory begins with the 1847 pa per of Reverand T. P. Kirkman "On a problem of Combinatorics," Cambridge and Dublin Math. Journal. The great Statistician R. A. Fisher reinvented the concept of combinatorial 2-design in the twentieth century. Extensive application of alge braic structures for construction of 2-designs (balanced incomplete block designs) can be found in R. C. Bose's 1939 Annals of Eugenics paper, "On the construction of balanced incomplete block designs." Coding Theory and Design Theory are closely interconnected. Hamming codes can be found (in disguise) in R. C. Bose's 1947 Sankhya paper "Mathematical theory of the symmetrical factorial designs.""

Spis treści

- Part I.- Baer subplanes, ovals and unitals.- On the length of codes with a given covering radius.- The differential encoding of coset codes by algebraic methods.- Families of codes with few distinct weights from singular and non-singular Hermitian varieties and quadrics in projective geometries and Hadamard difference sets and designs associated with two-weight codes.- Perfect multiple coverings in metric schemes.- Nonlinear feedforward sequences of m-sequences II.- Loops of clutters.- Positive independence and enumeration of codes with a given distance pattern.- Bounds on the number of pairs of unjoined points in a partial plane.- Inside Euclid’s algorithm.- Construction of designs.- Algebraic geometric codes.- Combinatorial characters of quasigroups.- Self-dual codes and self-dual designs.- The incidence algebra of a uniform poset.- Some recent results on signed graphs with least eigenvalues ? ?2.- Self-orthogonal codes and the topology of spinor groups.