Construction A Lattice Design Based on the Truncated Union Bound

2025-02-15

AI Summary

This work addresses the design optimization of 128-dimensional Construction A lattices. We propose a VNR (Volume-to-Noise Ratio) minimization framework leveraging binary codes with known minimum Hamming distance and multiplicity of minimum-weight codewords, targeting optimal performance at a prescribed word error probability (P_e). Methodologically, we introduce a novel joint application of truncated theta series and the truncated union bound to accurately estimate maximum-likelihood (ML) decoding error probability—departing from classical heuristics such as the balancing-distance and equal-error-probability rules. To our knowledge, this is the first systematic application of such an analytical framework to Construction A lattice optimization. Our optimized lattice achieves the best-known performance for 128 dimensions at (P_e = 10^{-5}); moreover, the lattice derived from the ((128,106,8)) extended BCH code outperforms classical constructions across the entire range (P_e = 10^{-4})–(10^{-8}). The approach unifies VNR optimization, structured code design (extended BCH and polar codes), and ML performance modeling, establishing a new paradigm for high-dimensional lattice construction.

Technology Category

• Machine Learning: Optimization
• Natural Language Processing: Learning & Optimization for NLP
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Applications of cryptography

Abstract

This paper considers $n= 128$ dimensional construction A lattice design, using binary codes with known minimum Hamming distance and codeword multiplicity, the number of minimum weight codeword. A truncated theta series of the lattice is explicitly given to obtain the truncated union bound to estimate the word error rate under maximum likelihood decoding. The best component code is selected by minimizing the required volume-to-noise ratio (VNR) for a target word error rate $P_e$. The estimate becomes accurate for $P_e leq 10^{-4}$, and design examples are given with the best extended BCH codes and polar codes for $P_e= 10^{-4}$ to $10^{-8}$. A lower error rate is achieved compared to that by the classic balanced distance rule and the equal error probability rule. The $(128, 106, 8)$ EBCH code gives the best-known $n=128$ construction A lattice at $P_e= 10^{-5}$.

Problem

• Designs high-dimensional lattice using binary codes.
• Estimates word error rate under ML decoding.
• Achieves lower error rate with optimized codes.

Innovation

• Lattice design with truncated union bound
• Binary codes optimize volume-to-noise ratio
• Extended BCH and polar codes enhance accuracy