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

TL;DR

This paper introduces the Voronoi spherical CDF $g_L(r)$ to capture the geometry of Voronoi cells for lattices and linear codes, and studies the normalized second moment and error resilience under Gaussian and Bernoulli noise. Using a first-moment argument together with Jensen's inequality and Siegel's summation formula, it derives non-asymptotic, dimension-free bounds on the average Voronoi-CDF and translates these into tight bounds on NSM and error probability for random lattices, as well as on Hamming distortion and error probability for random linear codes. The results show that, for most lattices in high dimensions, the NSM is within a factor $1+O(1/n)$ of the ball's NSM, and that Zador's bound for non-lattice quantizers can be approached by lattices up to an exponentially small additive term; similarly, linear codes achieve a distortion within a universal constant of the ball-based bound, and new finite-blocklength bounds near capacity improve upon previous bounds in the BSC/AWGN settings. The findings provide both tight non-asymptotic performance guarantees and evidence toward Gersho-type optimality for high-dimensional lattice quantizers, with concrete implications for quantization and reliable communication using lattices and linear codes.

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 trivial 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.

Figures (4)

• Figure 1: Various bounds and exact expressions for the the Voronoi Spherical CDF. For dimension $n=4$ we plot $g_{\mathcal{B}}(r)$ from \ref{['eq:gBall']}, the lower bound on ${\mathbb{E}}_{\mu_n}[g_L(r)]$ from \ref{['eq:gLjesnenbound']}, as well as the exact Voronoi Spherical CDF for $L={\mathbb{Z}}^{4}$. We also indicate the points ${r_{\mathrm{pack}}}({\mathbb{Z}}^4)$ where $g_{{\mathbb{Z}}^4}(r)$ and $g_{\mathcal{B}}(r)$ diverge, ${r_{\mathrm{cov}}}({\mathbb{Z}}^4)$ where $g_{{\mathbb{Z}}^{4}}(r)=1$ for the first time, and ${r_{\mathrm{eff}}}$ which is common to all unit covolume lattices, and this is where $g_{\mathcal{B}}(r)=1$ for the first time. For dimension $n=40$. We plot $g_{\mathcal{B}}(r)$ from \ref{['eq:gBall']}, the lower bound on ${\mathbb{E}}_{\mu_n}[g_L(r)]$ from \ref{['eq:gLjesnenbound']}, as well as the exact Voronoi Spherical CDF for $L=E_8^{\otimes 5}$ ($5$ copies of $E_8$) and $L={\mathbb{Z}}^{40}$. It is seen that our bound from \ref{['eq:gLjesnenbound']} is not very effective for $n=4$, but is already quite effective for $n=40$.
• Figure 2: Various bounds on the optimal NSM (due to agrell2025personal.
• Figure 3: Computation of the sphere packing lower bound, the $P_e^{\mathrm{MLB}}(\sigma^2)$ bound from izf12*Theorem 2, and the new upper bound from Theorem \ref{['thm:latticeSPUB2']}. The bounds are plotted for $\sigma^2=\frac{0.95}{2\pi e}$ and $\sigma^2=\frac{0.98}{2\pi e}$.
• Figure 4: Computation of the sphere packing lower bound, the RCU bound from ppv10*Theorem 33, and the new upper bound from Theorem \ref{['thm:PeRCUB']}. The bounds are plotted for $p=0.07$ and $p=0.1$, even values of $n$ and $k=n/2$.

Theorems & Definitions (48)

• Proposition 2.1
• Proposition 2.2
• Theorem 2.3
• proof : Proof of Theorem \ref{['thm:Jensen']}
• Theorem 2.4
• Theorem 2.5
• proof
• Lemma 2.6
• proof
• Corollary 2.7