Algebraic space–time coding—a powerful technique developed in the context of multiple-input multiple-output (MIMO) wireless communications—has profited tremendously from tools from Class Field Theory and, more concretely, the theory of central simple algebras and their orders. During the last decade, the study of space–time codes for practical applications, and more recently for future generation (5G\(+\)) wireless systems, has provided a practical motivation for the consideration of many interesting mathematical problems. One such problem is the explicit computation of orders of central simple algebras with small discriminants. In this article, we consider the most interesting asymmetric MIMO channel setups and, for each treated case, we provide explicit pairs of fields and a corresponding non-norm element giving rise to a cyclic division algebra whose natural order has the minimum possible discriminant.
Central simple algebras Division algebras Discriminant Natural orders MIMO Space–time coding
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A. Barreal and C. Hollanti are financially supported by the Academy of Finland Grants #276031, #282938, and #303819, as well as a grant from the Foundation for Aalto University Science and Technology. The authors thank Jean Martinet, René Schoof, and Bharath Sethuraman for their useful suggestions, and the anonymous reviewers for their valuable comments to improve the quality of the manuscript.
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