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On Fast-Decodable Algebraic Space–Time Codes

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Number Theory Meets Wireless Communications

Part of the book series: Mathematical Engineering ((MATHENGIN))

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Abstract

In the near future, the 5th generation (5G) of wireless systems will be well established. They will consist of an integration of different techniques, including distributed antenna systems and massive multiple-input multiple-output (MIMO) systems, and the overall performance will highly depend on the channel coding techniques employed. Due to the nature of future wireless networks, space–time codes are no longer merely an object of choice, but will often appear naturally in the communications setting. However, as the involved communication devices often exhibit a modest computational power, the complexity of the codes to be utilised should be reasonably low for possible practical implementation. Fast-decodable codes enjoy reduced complexity of maximum-likelihood (ML) decoding due to a smart inner structure allowing for parallelisation in the ML search. The complexity reductions considered in this chapter are entirely owing to the algebraic structure of the considered codes, and could be further improved by employing non-ML decoding methods, however yielding suboptimal performance. The aim of this chapter is twofold. First, we provide a tutorial introduction to space–time coding and study powerful algebraic tools for their design and construction. Secondly, we revisit algebraic techniques used for reducing the worst-case decoding complexity of both single-user and multiuser space-time codes, alongside with general code families and illustrative examples.

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Notes

  1. 1.

    By discrete, we mean that the metric on \(\mathbb {R}^n\) defines the discrete topology on Λ, i.e., any bounded region of \(\mathbb {R}^n\) contains only finitely many points of the subgroup.

  2. 2.

    The practical reason behind this choice is that the popular modulation alphabets, referred to as pulse amplitude modulation (PAM) and quadrature amplitude modulation (QAM), correspond to the rings of integers of these fields.

  3. 3.

    The smallest Frobenius norms correspond to the shortest Euclidean norms of the vectorised matrices. Directly, this would mean spherical constellation shaping. However, it is often more practical to consider a slightly more relaxed cubic shaping. This is the case in particular when the lattice in question is orthogonal, as then the so-called Gray-mapping [13] will give an optimal bit labelling of the lattice points.

  4. 4.

    In the literature, the code rate is often defined in complex symbols per channel use. We prefer using real symbols, as not every code admits a Gaussian basis, while every lattice has a \(\mathbb {Z}\)-basis.

  5. 5.

    This is a usual trick to balance the average energies of the codeword entries more evenly. See [3, Ex. 1] for more details.

  6. 6.

    As remarked in Sect. 3.4.1, the property of fast decodability is independent of the channel. Hence, we omit details on the structure of the effective channel.

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Authors and Affiliations

  1. Department of Mathematics and Systems Analysis, Aalto University, Aalto, Finland

    Amaro Barreal & Camilla Hollanti

Authors
  1. Amaro Barreal
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  2. Camilla Hollanti
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Correspondence to Camilla Hollanti .

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Editors and Affiliations

  1. Department of Mathematics, University of York, York, UK

    Victor Beresnevich

  2. Department of Electronic Engineering, University of York, York, UK

    Alister Burr

  3. Department of Electrical and Computer Engineering, Boston University, Boston, MA, USA

    Bobak Nazer

  4. Department of Mathematics, University of York, York, UK

    Sanju Velani

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Barreal, A., Hollanti, C. (2020). On Fast-Decodable Algebraic Space–Time Codes. In: Beresnevich, V., Burr, A., Nazer, B., Velani, S. (eds) Number Theory Meets Wireless Communications. Mathematical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-030-61303-7_3

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  • DOI: https://doi.org/10.1007/978-3-030-61303-7_3

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