Codes arising from algebraic structures over number fields lead naturally to determinant optimization problems governed by arithmetic invariants. In this paper, we investigate $2\times 2$ space-time block codes defined over rings of integers of imaginary quadratic fields, combining tools from algebraic number theory, cyclic algebras, and lattice theory. We prove that the Eisenstein construction over $\mathbb{Z}[ζ_3]$ is optimal within the family considered here: it attains the largest normalized density among the $2\times 2$ space-time block codes arising from rings of integers of imaginary quadratic fields. As a first step, we show that any code that could improve upon the Eisenstein construction must be defined over the ring of integers of $\mathbb{Q}(\sqrt{-d})$ with $d\in\{2,7,11\}$, apart from the classical Gaussian and Eisenstein cases. We then analyze these remaining fields by explicit arithmetic arguments, determine the optimal constructions over them, and show that none of them improves upon the Eisenstein code. A key ingredient in our approach is the derivation of effective non-norm criteria for quadratic extensions of imaginary quadratic fields. These criteria are obtained by local methods involving $2$-adic and $3$-adic valuations together with Hensel's lemma, and they ensure the division algebra property required for full diversity. They may also be of independent interest in the study of division algebras and their applications to coding theory and lattice-based communication.

翻译：由数域上代数结构导出的码自然引出了由算术不变量控制的行列式优化问题。本文研究定义于虚二次域整数环上的$2\times 2$空时分组码，综合运用代数数论、循环代数与格论的工具。我们证明：在本文考虑的族内，$\mathbb{Z}[ζ_3]$上的艾森斯坦构造是最优的——它在源于虚二次域整数环的$2\times 2$空时分组码中实现了最大正规化密度。作为第一步，我们证明：任何可能改进艾森斯坦构造的码必须定义在$\mathbb{Q}(\sqrt{-d})$（$d\in\{2,7,11\}$）的整数环上（经典高斯情形与艾森斯坦情形除外）。随后，我们通过显式算术论证分析这些剩余数域，确定其上的最优构造，并证明其中没有一种能改进艾森斯坦码。我们方法的一个关键要素是导出虚二次域二次扩张的有效非范数准则。这些准则通过涉及$2$进和$3$进赋值与亨泽尔引理的局部方法获得，并确保了全分集所需的除环性质。它们也可能在除环研究及其在编码理论与基于格的通信中的应用中具有独立价值。