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

2025-06-24

AI Summary

This work addresses performance analysis of random lattices and linear codes over additive white Gaussian noise (AWGN) and binary symmetric channels (BSC). To characterize the geometric structure of decoding regions, we introduce, for the first time, the Voronoi spherical cumulative distribution function (CDF). Using first-moment analysis combined with Jensen’s inequality, and leveraging uniform distribution modeling and geometric probability, we derive a tight analytical lower bound on the expected Voronoi spherical CDF. This bound substantially improves upon conventional spherical-region assumptions, yielding tighter upper bounds on the normalized second moment, bit error rate, and Hamming distortion. The key contribution lies in systematically incorporating the stochastic geometric structure of Voronoi cells into coding-theoretic performance analysis—establishing a novel analytical tool and delivering more precise theoretical limits for lattice quantization and linear coding.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Evaluation and Analysis
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Social media analysis through the lenses of networks

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 quite close to a trivial ball-based lower bound and immediately translate to improved 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 improved upper bounds on the Hamming distortion and the error probability of a random linear code over the binary symmetric channel.

Problem

• Define Voronoi spherical CDF for lattices/codes
• Develop lower bounds on expected CDF
• Improve bounds on coding performance metrics

Innovation

• Voronoi spherical CDF for lattices
• First moment method with Jensen's inequality
• Improved bounds on coding performance