On average coherence of cyclotomic lattices

TL;DR

This paper introduces the concepts of maximal and average coherence for lattices, investigates these properties in cyclotomic lattices, and compares their coherence to standard root lattices, with implications for signal processing and sphere packing.

Contribution

It defines and analyzes average coherence in lattices derived from cyclotomic number fields, providing a simple formula and comparative insights.

Findings

Derived a formula for average coherence of cyclotomic lattices

Compared coherence properties with standard root lattices

Highlighted potential applications in signal processing and sphere packing

Abstract

We introduce maximal and average coherence on lattices by analogy with these notions on frames in Euclidean spaces. Lattices with low coherence can be of interest in signal processing, whereas lattices with high orthogonality defect are of interest in sphere packing problems. As such, coherence and orthogonality defect are different measures of the extent to which a lattice fails to be orthogonal, and maximizing their quotient (normalized for the number of minimal vectors with respect to dimension) gives lattices with particularly good optimization properties. While orthogonality defect is a fairly classical and well-studied notion on various families of lattices, coherence is not. We investigate coherence properties of a nice family of algebraic lattices coming from rings of integers in cyclotomic number fields, proving a simple formula for their average coherence. We look at some examples of such lattices and compare their coherence properties to those of the standard root lattices.