Constructive Spherical Codes on Layers of Flat Tori

A new class of spherical codes is constructed by selecting a finite subset of flat tori from a foliation of the unit sphere S^2L-1 ⊂ R^2L and designing a structured codebook on each torus layer. The resulting spherical code can be the image of a lattice restricted to a specific box in R^L in each layer. Group structure and homogeneity, useful for efficient storage and decoding, are inherited from the underlying lattice codebook. A systematic method for constructing such codes are presented as well some examples of constructions. Upper and lower bounds on the performance, the asymptotic packing density and a method for decoding are derived.