The Voronoi Region of the Barnes-Wall Lattice $\Lambda_{16}$

TL;DR

This paper provides a detailed geometric and algebraic analysis of the Voronoi region of the Barnes-Wall lattice Λ₁₆, including exact quantizer constant calculation and novel variance estimation methods.

Contribution

It offers the first exact calculation of the Voronoi region's quantizer constant for Λ₁₆ and introduces a new simple, more accurate variance estimation technique.

Findings

Exact quantizer constant for Λ₁₆ obtained

Vertices and symmetry group of the Voronoi region characterized

New variance estimation method demonstrated

Abstract

We give a detailed description of the Voronoi region of the Barnes-Wall lattice $\Lambda_{16}$, including its vertices, relevant vectors, and symmetry group. The exact value of its quantizer constant is calculated, which was previously only known approximately. To verify the result, we estimate the same constant numerically and propose a new very simple method to quantify the variance of such estimates, which is far more accurate than the commonly used jackknife estimator.