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.

This work presents an extension of the Construction πAlattices proposed by Huang and Narayanan, 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 analyse the performance of the resulting multilevel lattice codes and highlight via computer simulations their notably reduced computational complexity provided by the multistage decoding. Moreover, it is shown that there is a sequence of Construction πAlattices that attain with high probability the Poltyrev-limit.

In this work we extend the Construction$\pi_{A}$lattices proposed in [1], 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 and show via computer simulations their notably reduced computational complexity provided by the multistage decoding.