# Non-norm criteria and optimal $2\times 2$ space-time block codes over rings of integers of imaginary quadratic fields

## TL;DR

This paper investigates optimal $2\times 2$ space-time block codes over rings of integers of imaginary quadratic fields, proving the Eisenstein construction over $\mathbb{Z}[\zeta_3]$ is optimal among such codes.

## Contribution

It establishes the optimality of the Eisenstein code within this family and rules out improvements over other quadratic fields using explicit arithmetic analysis.

## Key findings

- Eisenstein construction attains the largest normalized density among considered codes.
- No other codes over rings of integers of $\mathbb{Q}(\sqrt{-d})$ with $d\in\{2,7,11\}$ outperform the Eisenstein code.
- Derived effective non-norm criteria using local methods for quadratic extensions of imaginary quadratic fields.

## Abstract

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 $\mathbb{Z}[\zeta_3]$ 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.