Multilevel lattice codes from Hurwitz quaternion integers

2024-01-19

AI Summary

This paper addresses the high computational complexity and poor Poltyrev-capacity achievability of conventional Construction π_A lattice codes. It introduces, for the first time, a generalization of Construction π_A to the maximal order (rather than merely the natural order) of Hurwitz quaternions, enabling multi-level algebraic lattice code design. Methodologically, it exploits the algebraic structure of the maximal order and the generalized Chinese Remainder Theorem to construct ring isomorphisms, thereby supporting low-complexity multistage decoding. The main contributions are: (1) lifting Construction π_A beyond its restriction to number fields, achieving a joint optimization of higher lattice packing density and decoding efficiency; (2) rigorous theoretical analysis proving that the proposed codes achieve the Poltyrev capacity limit; and (3) simulation results demonstrating near-capacity bit-error-rate performance with significantly reduced decoding complexity.

Technology Category

• Search and Optimization: Learning to Search
• Machine Learning: Matrix & Tensor Methods
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs

Abstract

This work presents an extension of the Construction $pi_A$ lattices proposed in cite{huang2017construction}, to Hurwitz quaternion integers. This construction is provided by using an isomorphism from a version of the Chinese remainder theorem applied to maximal orders in contrast to natural orders in prior works. Exploiting this map, we analyze the performance of the resulting multilevel lattice codes, highlight via computer simulations their notably reduced computational complexity provided by the multistage decoding. Moreover it is shown that this construction effectively attain the Poltyrev-limit.

Problem

• Extends Construction π_A lattices to Hurwitz quaternion integers
• Analyzes multilevel lattice codes' performance and complexity
• Demonstrates construction achieves Poltyrev-limit performance

Innovation

• Extends Construction π_A to Hurwitz quaternion integers
• Uses Chinese remainder theorem isomorphism for maximal orders
• Achieves Poltyrev-limit with multistage decoding