Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
‘Time instances’ are commonly referred to as channel uses.
The noise is a combination of thermal noise and noise caused by the signal impulse.
From a mathematical point of view, any imaginary quadratic number field would give a nonvanishing determinant, but the choice \(\mathbb {Q}(i)\) matches with the quadrature amplitude modulation (QAM) commonly used in engineering.
This definition relates to the fact that a symmetric code carries the maximum amount of information (i.e., dimensions) that can be transmitted over a symmetric channel without causing accumulation points at the receiving end. In an asymmetric channel, a symmetric code will result in accumulation points, and hence asymmetric codes, i.e., non-full lattices are called for. See [9] for more details.
Having too few receive antennas will cause the lattice to collapse resulting in accumulation points, since the received signal now has dimension \(2n_rn_t<2n_t^2\). Hence, partial brute-force decoding of high complexity has to be carried out.
The factor \(\mathfrak p_{13}^5\) comes from the fact that \(E/L_3\) is tamely ramified and, thus, has as discriminant the prime \(\mathfrak q_{13}\) lying above 13 in \(L_3\).
References
Tarokh, V., Seshadri, N., Calderbank, A.R.: Space–time codes for high data rate wireless communication: performance criterion and code construction. IEEE Trans. Inf. Theory 44(2), 744–765 (1998)
Belfiore, J.-C., Rekaya, G.: Quaternionic lattices for space–time coding. In: Proceedings of the IEEE information theory workshop (2003)
Sethuraman, B.A., Rajan, B.S., Shashidhar, V.: Full-diversity, high-rate space–time block codes from division algebras. IEEE Trans. Inf. Theory 49(10), 2596–2616 (2003)
Belfiore, J.-C., Rekaya, G., Viterbo, E.: The golden code: a \(2\times 2\) full-rate space–time code with non-vanishing determinants. IEEE Trans. Inf. Theory 51(4), 1432–1436 (2005)
Oggier, F., Rekaya, G., Belfiore, J.-C., Viterbo, E.: Perfect space–time block codes. IEEE Trans. Inf. Theory 52(9), 3885–3902 (2006)
Elia, P., Sethuraman, B.A., Kumar, P.V.: Perfect space–time codes for any number of antennas. IEEE Trans. Inf. Theory 53(11), 3853–3868 (2007)
Hollanti, C., Lahtonen, J., Lu, H.-F.: Maximal orders in the design of dense space–time lattice codes. IEEE Trans. Inf. Theory 54(10), 4493–4510 (2008)
Vehkalahti, R., Hollanti, C., Lahtonen, J., Ranto, K.: On the densest MIMO lattices from cyclic division algebras. IEEE Trans. Inf. Theory 55(8), 3751–3780 (2009)
Hollanti, C., Lu, H.-F.: Construction methods for asymmetric and multiblock space–time codes. IEEE Trans. Inf. Theory 55(3), 1086–1103 (2009)
Reiner, I.: Maximal Orders. London Mathematical Society Monographs New Series, vol. 28. Oxford University Press, Oxford (2003)
Vehkalahti, R.: Class field theoretic methods in the design of lattice signal constellations, vol. 100. TUCS dissertations (2008)
Vehkalahti, R., Hollanti, C., Oggier, F.: Fast-decodable asymmetric space–time codes from division algebras. IEEE Trans. Inf. Theory 58(4), 2362–2385 (2012)
Cohen, H., Stevenhagen, P.: Computational class field theory. Algorithmic Number Theory 44, 497–534 (2008)
Milne, J.S.: Algebraic Number Theory (v3.07), Graduate Course Notes. www.jmilne.org/math/CourseNotes/ (2017)
Milne, J.S.: Class Field Theory (v4.02), Graduate Course Notes. www.jmilne.org/math/CourseNotes/ (2013)
Hudson, R.H., Williams, K.S.: The integers of a cyclic quartic field. Rocky Mt. J. Math. 20(1), 145–150 (1990)
Huard, J.G., Spearman, B.K., Williams, K.S.: Integral bases for quartic fields wit quadratic subfields. Carlet. Ott. Math. Lect. Notes Ser. 7, 87–102 (1986)
Bergé, A.-M., Martinet, J., Olivier, M.: The computation of sextic fields with a quadratic subfield. Math. Comput. 54(190), 869–884 (1990)
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Barreal, A., Corrales Rodrigáñez, C. & Hollanti, C. Natural orders for asymmetric space–time coding: minimizing the discriminant. AAECC 29, 371–391 (2018). https://doi.org/10.1007/s00200-017-0348-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00200-017-0348-5