On upper bounds for code distance and covering radius of designs in polynomial metric spaces

The purpose of this paper is to present new upper bounds for code distance and covering radius of designs in arbitrary polynomial metric spaces. These bounds and the necessary and sufficient conditions of their attainability were obtained as the solution of an extremal problem for systems of orthogonal polynomials. For antipodal spaces the behaviour of the bounds in different asymptotical processes is determined and it is proved that this bound is attained for all tight 2k-design.