Commutative group codes in and —Approaching the bound

Spherical codes in even dimensions n = 2 m generated by a commutative group of orthogonal matrices can be determined by a quotient of m -dimensional lattices when the sublattice has an orthogonal basis. We discuss here the existence of orthogonal sublattices of the lattices A 2 , D 3 , D 4 and E 8 , which have the best packing density in their dimensions, in order to generate families of commutative group codes approaching the bound presented in Siqueira and Costa (2008)  [14].