[1901.06200] Non-norm criteria and optimal $2\times 2$ space-time block codes over rings of integers of imaginary quadratic fields Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Computer Science > Information Theory arXiv:1901.06200 (cs) [Submitted on 18 Jan 2019 ( v1 ), last revised 30 Apr 2026 (this version, v2)] Title: Non-norm criteria and optimal $2\times 2$ space-time block codes over rings of integers of imaginary quadratic fields Authors: Carina Alves , Eliton Mendonça Moro , Cintya Wink de Oliveira Benedito , Antonio Aparecido de Andrade View a PDF of the paper titled Non-norm criteria and optimal $2\times 2$ space-time block codes over rings of integers of imaginary quadratic fields, by Carina Alves and 2 other authors View PDF HTML (experimental) 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. Comments: 14 pages Subjects: Information Theory (cs.IT) MSC classes: 11R52, 11R04, 94B05 Cite as: arXiv:1901.06200 [cs.IT] (or arXiv:1901.06200v2 [cs.IT] for this version) https://doi.org/10.48550/arXiv.1901.06200 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Carina Alves [ view email ] [v1] Fri, 18 Jan 2019 12:22:30 UTC (19 KB) [v2] Thu, 30 Apr 2026 22:22:16 UTC (27 KB) Full-text links: Access Paper: View a PDF of the paper titled Non-norm criteria and optimal $2\times 2$ space-time block codes over rings of integers of imaginary quadratic fields, by Carina Alves and 2 other authors View PDF HTML (experimental) TeX Source view license Current browse context: cs.IT < prev | next > new | recent | 2019-01 Change to browse by: cs math math.IT References & Citations NASA ADS Google Scholar Semantic Scholar DBLP - CS Bibliography listing | bibtex Carina Alves Eliton M. Moro export BibTeX citation Loading... BibTeX formatted citation × loading... 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