Mathematics of Digital Communications: From Finite Fields to Group Rings and Noncommutative Algebra
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REVIEW PAPER
Mathematics of Digital Communications: From Finite Fields to Group Rings and Noncommutative Algebra Hassan Khodaiemehr1,2 • Dariush Kiani2,3 Received: 31 October 2019 / Accepted: 6 January 2020 Ó Shiraz University 2020
Abstract In this paper, we present some recent instances of applying algebraic tools from different categories, in the design of digital communications systems. More specifically, we present a structure based on the elements of a group ring for constructing and encoding QC-LDPC codes. As another instance, we present the construction of space-time block codes from the rings of twisted Laurent series which include some instances of crossed product and non-crossed product division algebras. Keywords Crossed product algebra Group ring LDPC codes Mathematics Subject Classification 16S35 12E15 94B05
1 Introduction Algebraic tools are often proved to be useful in places where no one could expect them at all. Some instances of these applications started by applying elementary number theory in the development of error correcting codes in the early years of coding theory and after that, finite fields became the key tools in the mathematical background of digital communications and in the design of powerful binary codes. In the recent years, other algebraic structures like group rings are also applied to improve the efficiency and performance of binary codes (Khodaiemehr and Kiani 2017). Due to the technological developments and increased processing power of digital receivers, attention moved to the design of signal space codes in the framework of coded modulation systems. Here, the theory of & Dariush Kiani [email protected] Hassan Khodaiemehr [email protected] 1
Department of Mathematics, K. N. Toosi University of Technology, Tehran 16315-1618, Iran
2
School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran 19395-5746, Iran
3
Department of Mathematics and Computer Science, Amirkabir University of Technology, Tehran 15875-4413, Iran
Euclidean lattices, which are discrete additive groups of Rn , became of great interest for the design of dense signal constellations well suited for transmission over the additive white Gaussian noise (AWGN) channel. More recently, the incredible developments of wireless communications forced coding theorists to deal with fading channels. New code design criteria had to be considered in order to improve the poor performance of wireless transmission systems. The need for bandwidth-efficient coded modulation became even more important due to scarce availability of radio bands (Oggier and Viterbo 2004; Khodaiemehr et al. 2015, 2016b, 2017a, b; Khodaiemehr and Eghlidos 2018). Algebraic number theory was shown to be a very useful mathematical tool that enables the design of good coding schemes for fading channels. These codes are constructed as multidimensional lattice signal sets (or constellations) with particular geometric properties. Most of the coding gain is obtained by in
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