Non-norm criteria and optimal $2\times 2$ space-time block codes over rings of integers of imaginary quadratic fields
For coding theorists working on space-time block codes, this paper provides a definitive optimality result within a specific algebraic family, though the result is incremental as it confirms the Eisenstein construction's superiority over a limited set of alternatives.
This paper proves that the Eisenstein construction over Z[ζ3] yields the optimal 2×2 space-time block code among those defined over rings of integers of imaginary quadratic fields, achieving the largest normalized density. The authors show that any potentially better code must come from fields with d in {2,7,11}, and then demonstrate that none of these improves upon the Eisenstein code.
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.