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

Abstract

We give a detailed description of the Voronoi region of the Barnes-Wall lattice $Λ_{16}$, 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.

Figures (2)

• Figure 1: The $22$ geometrically distinct types of $2$-faces of the Voronoi region of $\Lambda_{16}$. We show the area $A$ as well as the number of classes the face types fall into under $\mathop{\mathrm{Aut}}\nolimits(\Lambda_{16})$. For reference, the line in the top-left corner has unit length. (a) is a square with edges of length $1/3$. The triangle (d) is equilateral, while (b), (c), (e), (g), (h), (l), (p), (t), and (v) are isosceles. All internal angles of the triangles are strictly less than $90^\circ$. The maximum $\theta_\text{max}$ is only found in (n), where $\cos\theta_\text{max} = \sqrt{10}/40$ ($\theta_\text{max} \approx 85.47^\circ$). The smallest angle satisfies $\cos\theta_\text{min} = 29\sqrt{238}/476$ ($\theta_\text{min} \approx 19.97^\circ$) and is only found in triangle (q).
• Figure 2: Histograms of two estimates of the standard deviation of the estimated second moment $\hat{U}$ of the cubic lattice. The exact standard deviation $(\mathop{\mathrm{var}}\nolimits \hat{U})^{1/2}$, which can be calculated analytically for the cubic lattice, reveals that the proposed estimator \ref{['eq:uhat']} is much more accurate than the jackknife with 100 groups.