Nonuniform Fuchsian codes for noisy channels

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Abstract

We develop a new transmission scheme for additive white Gaussian noisy (AWGN) channels based on Fuchsian groups from rational quaternion algebras. The structure of the proposed Fuchsian codes is nonlinear and nonuniform, hence conventional decoding methods based on linearity and symmetry do not apply. Previously, only brute force decoding methods with complexity that is linear in the code size exist for general nonuniform codes. However, the properly discontinuous character of the action of the Fuchsian groups on the complex upper half-plane translates into decoding complexity that is logarithmic in the code size via a recently introduced point reduction algorithm.

Introduction

Fuchsian groups constructed from quaternion algebras arise in the study of Shimura curves [18], a rich theory with a large number of theoretical applications to various branches of number theory like Jacquet–Langlands correspondence or the proof of the Shimura–Taniyama–Weil conjecture. Shimura curves are also present in the theory of error-correcting codes [11]. More recently, Fuchsian groups have made an appearance [19], [23], [6], [21] in the context of signal constellation design with potential applications in communications.

In this paper,3 we will consider a new family of Fuchsian codes. The codes are obtained from unit groups of orders of quaternion algebras acting on the complex upper half-plane, in this way giving rise to complex points that can be used as codewords. Each of the above notions will be properly introduced in the sequel, but let us first concentrate on the general communication problem at hand.

Namely, as the underlying mathematical communication model, we will use the typical additive white Gaussian noise (AWGN) channel model [8, Chapter 10]. The transmission process is described by the equation:y=x+w,where yC is the received signal, xC is the transmitted codeword drawn from a finite codebook CC (also referred to as a constellation), and w is complex AWGN with zero mean and variance σ2/2 per real and imaginary part.

Throughout this paper, we denote by R(z) and I(z) the real and imaginary part of a complex number zC, respectively. The complex absolute value, i.e., Euclidean norm is denoted by |z|=R(z)2+I(z)2, and the cardinality of a code C by |C|. In spite of the slight abuse of notation there should not be any danger of confusion.

Next, we summarize our main contributions and reflect our work to relevant earlier work related to Fuchsian groups in the context of communication applications. The main contributions of this paper are

  • We show how to explicitly build nonuniform signal constellations on the complex plane by using Fuchsian groups and Möbius transformations. Nonuniform signal constellations are included in the digital video broadcasting standard for next generation handheld (DVB-NGH) systems, and they are currently being considered for the future extension of terrestrial DVB with multiple antennas (DVB-T2 MIMO). This creates a great interest and need for nonuniform constellations.

  • We describe the whole encoding and decoding process of the proposed Fuchsian codes in full detail, assuming the AWGN communication setting.

  • Our construction method allows for decoding complexity which is logarithmic in the code size, enabled by the so-called point reduction algorithm [2] based on determining the tile to which a given point belongs in the hyperbolic upper half-plane. This is a magnificent improvement since, as far as the authors are aware, there are no known optimal decoders for general nonuniform constellations with sublinear complexity.

  • We also discuss the optimization of the Fuchsian codes and propose a new design criterion, hence motivating further study on Fuchsian codes.

  • Finally, we present an alternative method for constructing Fuchsian codes by certain parametrization of the integer tuples defining the Möbius transformations used for the code construction.

Our interest in Fuchsian groups as a basis for code construction stems from a series of recent papers by Palazzo et al. In [23], [6], [19], [21], among others, various interesting connections between Fuchsian groups and signal constellation design are presented. In [23], the authors construct Fuchsian groups suitable for signal constellation construction. In [19], the authors consider the unit disk model of the hyperbolic half-plane as the signal space, and the noise is modeled as a hyperbolic Gaussian random variable. With the study of the hyperbolic geometry they construct a hyperbolic equivalent to QAM and PSK constellations and point out that, when the channel model is hyperbolic,4 the proposed hyperbolic constellations provide higher coding gains than the classical euclidean variants. Building on this work, in [21] the authors construct dense tessellations and counting Dirichlet domains in tessellations of certain type. In [6] the authors use units of quaternion orders to construct space-time matrices with the potential use case being wireless multi-antenna (MIMO) communications. We refer the reader to [17], [14] as the early references to the use of division algebras and maximal orders in MIMO, and to [3] for a more general introduction to the topic.

Although codes related to Fuchsian groups have been considered before, our construction is original in that it describes the complete construction and decoding process, whereas earlier work has largely concentrated on the constellation design while giving little attention to the decoding and performance aspects. Another key difference to the aforementioned works is that we are studying codes on the complex plane arising from quaternion algebras and Fuchsian groups, and our aim is to apply the codes to the classical (euclidean) channel models such as the aforementioned AWGN channel, with possible future extension to fading channels [16], [3]. We do not use hyperbolic metric as our design metric, but use the Fuchsian group as a starting point to the code generation. Nevertheless, our decoder will rely on hyperbolic geometry as opposed to the classical decoders based on euclidean geometry.

The paper is organized as follows. In what remains of this section, we will give some insight to AWGN channel decoding. In Section 2 we provide the essential algebraic preliminaries. The Fuchsian code construction process as well as decoding via point reduction algorithm are introduced in Section 3. Section 4 provides a thorough decoding complexity analysis, showing that the decoding algorithm has logarithmic complexity. We discuss the optimization of the proposed Fuchsian codes in Section 5 as a motivation for further research. Conclusions and directions for further research are given in Section 6. Finally, we present as an appendix an alternative method for constructing Fuchsian codes. This method is called for when the generators of the Fuchsian group are not known.

Let us discuss the decoding process in AWGN channels before going to the actual code construction in more detail. This decoding process, i.e., deciding on which codeword xC was transmitted given the received signal yC can be done in many different ways. An optimal decoding method is given by the maximum-likelihood (ML) decoding, which decides on the codeword x^ having the smallest squared euclidean distance to y:x^=argmin|yx|2.This amounts to exhaustively enumerating the metric (1.2) for all xC, and comparing the values obtained in order to find the minimum. The metric evaluations require 4|C| arithmetic operations,5 and to compare, we have to compute |C|1 differences. In total, this amounts to 5|C|1 arithmetic operations. As far as the authors are aware, there are no other known optimal decoding methods for general nonuniform codes.

In [4], we have compared the error performance6 of some Fuchsian codes to that of quadrature amplitude modulation (QAM) in order to get some preliminary insight as to how close to these classical constellations we are able to get. We define an odd, symmetric square QAM constellation as 22rQAM={±a±bi|1a,b2r1,2ab}Z[i].This is a subset of the two-dimensional Gaussian integer lattice7 Z[i], hence its ML complexity can be written as 5|C|1=5|S|21, where SZ is the corresponding real pulse amplitude modulation (PAM) constellation: 2rPAM={(2r1),,3,1,1,3,,2r1}Z.More generally, if we denote by S the underlying real signaling alphabet Z of a lattice code, the ML complexity 5|C|1=5|S|κ1 grows exponentially with the lattice dimension κ.

For lattice codes, the ML complexity can be reduced by using lattice decoding, which performs a closest lattice point search within a limited sphere centered at the received point y, while ignoring the fact that the codebook is a finite subset of the infinite lattice. The complexity of lattice decoding is hence independent of |C|, and it actually turns out to be polynomial (cf. [25]) in |S| for a given lattice and sphere radius. Unfortunately it also performs poorly compared to ML decoding. The performance can be improved by taking into account the code boundaries, often referred to as sphere decoding, but this again increases the complexity. Naturally, the worst case complexity of a sphere decoder is always upper bounded by the complexity of exhaustive search.

The complexity comparison between the QAM constellations and Fuchsian constellations is not straightforward since, in practice, one does not use ML, lattice or sphere decoder for decoding QAM in the single-input single-output (SISO) case (cf. Eq. (1.1)). The difficulty of complexity comparison stems from the fact that, while the decoding complexity of the proposed Fuchsian codes largely arises from arithmetic operations, the decoding complexity of QAM in the SISO case is, in practice,8 a combination of arithmetic operations and memory usage due to maintenance of a look-up table. So for QAM, this finally boils down to resource usage in a particular chip, the trade-off being memory vs. arithmetic operations. In addition, the estimate quality of the received signal is a parameter, since the amount of memory depends on the bit-resolution of the look-up table. In the literature, a look-up table is normally hand-waved as having negligible complexity, whereas in reality a very large table could still be highly inconvenient. Due to this comparison mismatch, we compare the complexity of Fuchsian codes to the ML decoding complexity 5|C|1. This is also a more righteous comparison in the sense that, as noted before, nonuniform codes are not previously known to admit sublinear decoding complexity. Indeed, one of the main contributions of this paper is that our codes enable the use of a decoding algorithm with complexity that is logarithmic in the code size |C|.

Remark 1.1

We have chosen to use the number of arithmetic operations as the complexity measure. Another option would be to only count multiplications and divisions, since these are more complex than addition and subtraction. Nevertheless, both options yield very similar results. In addition, when the numbers involved in the arithmetic operations have known and predetermined precision, all arithmetic operations can be thought of as constant-time operations.

Section snippets

Algebraic preliminaries

In this section, we survey some facts on the arithmetic of quaternion algebras in order to construct a discrete group ΓSL(2,R) and its fundamental domain in the complex upper half-plane. We mainly follow [1] and refer the reader to the well-known references [15], [24] for more details.

Construction of Fuchsian codes

In this section, we will show in detail how to construct and decode Fuchsian codes in an AWGN channel. The first subsection describes our proposal for the construction of a new family of Fuchsian codes. The second subsection introduces the point reduction algorithm (PRA) [2], which will be used for decoding in the third subsection.

In what follows Γ will be a Fuchsian group Γ(D,1) with D>1 a product of an even number of primes. In fact, our construction could be formulated more generally in

Complexity

In this section we see how the properly discontinuous character of the action of a Fuchsian group Γ implies fast decoding. Let C={±γ(τ)γSΓ} be the codebook. Since we have chosen τ in the interior of F, all the points in the codebook are indeed distinct, so |C|=2N.

Consider G the set of elements in Γ defined in Section 2 according to the election of the fundamental domain, G=GintGext.

Proposition 4.1

The complexity of the decoding algorithm for a Fuchsian code C, in number of arithmetic operations (i.e. sums,

Code design criterion for Fuchsian code optimization

In the previous sections, we have shown how to construct Fuchsian codes from scratch and how to decode them with the point reduction algorithm (PRA). However, the simulations we have carried out (cf. [4]) demonstrate that the typical design criterion for codes used in conjunction with a ML (or lattice decoder) does not work for the PRA decoder. Hence, there is a call for a new design criterion for codes to be used in conjunction with PRA decoder.

In more detail, for ML decoding, the performance

Conclusions and further research

In this paper, we have designed a new class of codes called Fuchsian codes. These codes were obtained by considering constellations on the complex plane arising from the Möbius transformation related to a Fuchsian group coming from units in rational quaternion algebras.

We have described the construction and decoding process of the proposed codes in full detail, providing also numerous explicit examples. According to [4], the differences in the performance of different Fuchsian codes can vary

Acknowledgments

The authors gratefully acknowledge the support from the European Science Foundation׳s COST Action IC1104 and from the research project MTM2012-33830 (MICINN/UB, Spain), as well as the hospitality of the Institute of Mathematics at the University of Barcelona (IMUB). They would also like to thank Peter Moss from the British Broadcasting Corporation (BBC) Research & Development for fruitful discussions on AWGN channels, and Professor Pilar Bayer from University of Barcelona for sharing her

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