Finite-Blocklength Analysis of Alamouti Codes over Eisenstein Integers

2026-04-11

AI Summary

This work addresses the energy efficiency and reliability challenges of short-packet communications with Alamouti codes by proposing a novel Alamouti–Eisenstein code constructed from the maximal order of the quaternion algebra $(-1,-3)_\mathbb{Q}$. For the first time, Eisenstein integers are incorporated into Alamouti code design, yielding an $A_2 \oplus A_2$ lattice structure and hexagonal shaping gain. Under complex embedding, the code achieves full diversity, orthogonality, and non-vanishing determinants. Leveraging tools from algebraic number theory, lattice theory, and finite-blocklength information theory, the proposed scheme significantly enhances reliability for short-packet transmission compared to conventional Gaussian-integer-based Alamouti codes at the same code rate and signal-to-noise ratio, offering an asymptotic energy gain of approximately 0.79 dB and a modest improvement in mutual information.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Multiagent Systems: Agent Communication
• Intelligent Robots: State Estimation

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Energy management for devices in mobile Web and WoT environments
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

We study a space--time block code from a maximal order in the definite quaternion algebra $(-1,-3)_{\Q}$. Its embedding into $\C^{2\times 2}$ yields an Alamouti--Eisenstein code over $\Z[w]$ with full diversity, orthogonality, and non-vanishing determinant. The underlying lattice is isomorphic to $\Z[w]^2$, while the embedded lattice has $A_2\oplus A_2$ geometry, yielding a hexagonal shaping gain. We compare it with the classical Alamouti code over $\Z[i]$ in terms of shaping, constellation-constrained mutual information, and finite-blocklength achievable rates, obtaining an asymptotic energy gain of about $0.79$~dB and a small but positive mutual-information gain. At the same SNR and rate, the Alamouti--Eisenstein design also improves short-packet reliability.

Problem

• finite-blocklength
• Alamouti codes
• Eisenstein integers
• short-packet reliability
• space-time block coding

Innovation

• Alamouti-Eisenstein code
• finite-blocklength analysis
• hexagonal shaping gain
• non-vanishing determinant
• space-time block coding