Finite-Blocklength Analysis of Alamouti Codes over Eisenstein Integers

We study a space–time block code from a maximal order in the definite quaternion algebra $(-1,-3{)}_{\text{Q}}$. Its embedding into ${\text{C}}^{2\times 2}$ yields an Alamouti–Eisenstein code over $\mathbb{Z}[w]$ with full diversity, orthogonality, and non-vanishing determinant. The underlying lattice is isomorphic to $\mathbb{Z}[w{]}^2$, 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 $\mathbb{Z}[i]$ 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.