Bounds for codes via semidefinite programming

Delsarte´s method and its extensions allow to consider the upper bound problem for codes in 2-point-homogeneous spaces as a linear programming problem with perhaps infinitely many variables. In this paper two extensions of Delsarte´s method via semidefinite programming are considered. The first approach shows that using as variables power sums of distances this problem can be considered as a finite semidefinite programming problem. This method allows to improve some upper bounds. The second approach extends the Bachoc-Vallentin method for spherical codes. In particular, an extension of Schoenberg´s theorem for multivariate Gegenbauer polynomials has been proved.