Commutative group codes in R4,R6,R8R4,R6,R8 and R16R16—Approaching the bound

Spherical codes in even dimensions n=2mn=2m generated by a commutative group of orthogonal matrices can be determined by a quotient of mm-dimensional lattices when the sublattice has an orthogonal basis. We discuss here the existence of orthogonal sublattices of the lattices A2,D3,D4 and E8E8, 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].