Finite-Blocklength Analysis of Alamouti Codes over Eisenstein Integers

TL;DR

This paper introduces a novel Alamouti--Eisenstein space--time code over Eisenstein integers, demonstrating improved energy efficiency and reliability over classical codes through finite-blocklength analysis.

Contribution

It presents a new space--time code based on Eisenstein integers with full diversity and improved performance metrics compared to classical codes.

Findings

Achieves about 0.79 dB energy gain over classical Alamouti code.

Improves mutual information and short-packet reliability at the same SNR and rate.

Exhibits hexagonal shaping gain due to lattice geometry.

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.