Krawtchouk Polynomials and Universal Bounds for Codes and Designs in Hamming Spaces

Abstract

Universal bounds for the cardinality of codes in the Hamming space Frn with a given minimum distance d and/or dual distance d′ are stated. A self-contained proof of optimality of these bounds in the framework of the linear programming method is given. The necessary and sufficient conditions for attainability of the bounds are found. The parameters of codes satisfying these conditions are presented in Table I. As consequences, in particular, a new upper bound for the minimum distance of self-dual codes and a new lower bound for the crosscorrelation of half-linear codes are obtained. © 1995 IEEE.