The Voronoi Spherical CDF for Lattices and Linear Codes: New Bounds for Quantization and Coding

TL;DR

This paper introduces the Voronoi spherical CDF for lattices and linear codes, providing new bounds on quantization and coding performance that are valid in any dimension and close to ideal bounds.

Contribution

It develops universal lower bounds on the Voronoi spherical CDF, leading to new non-asymptotic bounds on second moment, error probability, and Hamming distortion for lattices and codes.

Findings

Most lattices have second moments close to Euclidean balls with same covolume.

Expected Hamming distortion of most linear codes is close to that of Hamming balls.

Bounds are valid for any finite dimension and nearly tight.

Abstract

For a lattice/linear code, we define the Voronoi spherical cumulative density function (CDF) as the CDF of the $\ell_2$-norm/Hamming weight of a random vector uniformly distributed over the Voronoi cell. Using the first moment method together with a simple application of Jensen's inequality, we develop lower bounds on the expected Voronoi spherical CDF of a random lattice/linear code. Our bounds are valid for any finite dimension and are quite close to a ball-based lower bound. They immediately translate to new non-asymptotic upper bounds on the normalized second moment and the error probability of a random lattice over the additive white Gaussian noise channel, as well as new non-asymptotic upper bounds on the Hamming distortion and the error probability of a random linear code over the binary symmetric channel. In particular, we show that for most lattices in $\mathbb{R}^n$ the second moment is greater than that of a Euclidean ball with the same covolume only by a $\left(1+O(\frac{1}{n})\right)$ multiplicative factor. Similarly, for most linear codes in $\mathbb{F}_2^n$ the expected Hamming distortion is greater than that of a corresponding Hamming ball only by an additive universal constant.