Finite-Blocklength Analysis of Alamouti Codes over Eisenstein Integers
For wireless communication systems using Alamouti codes, this paper offers a modest incremental improvement in energy efficiency and finite-blocklength performance by replacing the Gaussian integer lattice with an Eisenstein integer lattice.
This work introduces an Alamouti-Eisenstein space-time block code over Eisenstein integers, achieving full diversity, orthogonality, and non-vanishing determinant with a hexagonal shaping gain. Compared to the classical Alamouti code over Gaussian integers, it provides an asymptotic energy gain of about 0.79 dB and improved short-packet reliability at the same SNR and rate.
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.