Constructive spherical codes on layers of flat tori

TL;DR

This paper introduces a new class of spherical codes constructed from flat tori layers on the sphere, leveraging lattice structures for efficient encoding, decoding, and performance analysis, exemplified by a code in R^{48} using the Leech lattice.

Contribution

It presents a systematic method for constructing spherical codes from lattice-restricted flat tori layers, combining geometric and algebraic structures for improved efficiency.

Findings

Constructed spherical codes using layered flat tori and lattice codes.

Derived bounds on code performance and asymptotic density.

Developed a decoding method exploiting group structure.

Abstract

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} of 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 hyperbox 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 and, as an example, the Leech lattice is used to construct a spherical code in R^{48}. Upper and lower bounds on the performance, the asymptotic packing density and a method for decoding are derived.