We prove that the kissing number in 48 dimensions among antipodal spherical codes with certain forbidden inner products is 52\,416\,000. Constructions of attaining codes as kissing configurations of minimum vectors in even unimodular extremal lattices are well known since the 1970's. We also prove that corresponding spherical 11-designs with the same cardinality are minimal. We use appropriate modifications of the linear programming bounds for spherical codes and designs introduced by Delsarte, Goethals and Seidel in 1977.