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Theory and Practice of Non-Binary Graph-Based Codes: A Combinatorial View

Behzad Amiri

Published Jan 1, 2015• 0 citations
OpenAlex

Abstract

We are undergoing a revolution in data. The ever-growing amount of information in our world has created an unprecedented demand for ultra-reliable, affordable, and resource-efficient data storage systems. Error-correcting codes, as a critical component of any memory device, will play a crucial role in the future of data storage.One particular class of error-correcting codes, known as graph-based codes, has drawn significant attention in both academia and in industry. Graph-based codes offer superior performance compared to traditional algebraic codes. Recently, it has been shown that non-binary graph-based codes, which operate over finite fields rather than binary alphabets, outperform their binary counterparts and exhibit outstanding overall performance. For this reason, these codes are particularly suitable for emerging data storage systems.In this dissertation, we present a comprehensive combinatorial analysis of non-binary graph-based codes. We perform both finite-length and asymptotic analyses for these codes, providing a systematic framework to evaluate and optimize various families of non-binary graph-based codes.In the finite-length case, we provide a mathematical characterization of the error floor problem, including a general definition of absorbing sets over non-binary alphabets. We consider several structured low-density parity-check (LDPC) codes, including quasi-cyclic and spatially-coupled codes, as well as unstructured LDPC codes. We offer design guidelines for non-binary LDPC codes with outstanding performance in extremely low error-rate regimes; making them excellent candidates for data storage applications.In the asymptotic case, we provide a novel toolbox for the evaluation of families of non-binary graph-based codes. By utilizing insights from graph theory and combinatorics, we establish enumerators for a general family of graph-based codes which are constructed based on protographs. We provide asymptotic distributions of codewords and trapping sets for the family of protograph-based codes. Furthermore, we present an asymptotic enumeration of binary and non-binary elementary absorbing sets for regular code ensembles.The contributions of this dissertation can potentially impact a broad range of data storage and communication technologies that require excellent performance in high-reliability regimes.

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We are undergoing a revolution in data. The ever-growing amount of information in our world has created an unprecedented demand for ultra-reliable, affordable, and resource-efficient data storage systems. Error-correcting codes, as a critical component of any memory device, will play a crucial role in the future of data storage.One particular class of error-correcting codes, known as graph-based codes, has drawn significant attention in both academia and in industry. Graph-based codes offer superior performance compared to traditional algebraic codes. Recently, it has been shown that non-binary graph-based codes, which operate over finite fields rather than binary alphabets, outperform their binary counterparts and exhibit outstanding overall performance. For this reason, these codes are particularly suitable for emerging data storage systems.In this dissertation, we present a comprehensive combinatorial analysis of non-binary graph-based codes. We perform both finite-length and asymptotic analyses for these codes, providing a systematic framework to evaluate and optimize various families of non-binary graph-based codes.In the finite-length case, we provide a mathematical characterization of the error floor problem, including a general definition of absorbing sets over non-binary alphabets. We consider several structured low-density parity-check (LDPC) codes, including quasi-cyclic and spatially-coupled codes, as well as unstructured LDPC codes. We offer design guidelines for non-binary LDPC codes with outstanding performance in extremely low error-rate regimes; making them excellent candidates for data storage applications.In the asymptotic case, we provide a novel toolbox for the evaluation of families of non-binary graph-based codes. By utilizing insights from graph theory and combinatorics, we establish enumerators for a general family of graph-based codes which are constructed based on protographs. We provide asymptotic distributions of codewords and trapping sets for the family of protograph-based codes. Furthermore, we present an asymptotic enumeration of binary and non-binary elementary absorbing sets for regular code ensembles.The contributions of this dissertation can potentially impact a broad range of data storage and communication technologies that require excellent performance in high-reliability regimes.
This paper provides a comprehensive analysis of nonbinary low-density parity check (LDPC) codes built out of protographs. We consider both random and constrained edge-weight labeling, and refer to the former as the unconstrained nonbinary protograph-based LDPC codes (U-NBPB codes) and to the latter as the constrained nonbinary protograph-based LDPC codes (C-NBPB codes). Equipped with combinatorial definitions extended to the nonbinary domain, ensemble enumerators of codewords, trapping sets, stopping sets, and pseudocodewords are calculated. The exact enumerators are presented in the finite-length regime, and the corresponding growth rates are calculated in the asymptotic regime. An EXIT chart tool for computing the iterative decoding thresholds of protograph-based LDPC codes is presented, followed by several examples of finite-length U-NBPB and C-NBPB codes with high performance. Throughout this paper, we provide accompanying examples, which demonstrate the advantage of nonbinary protograph-based LDPC codes over their binary counterparts and over random constructions. The results presented in this paper advance the analytical toolbox of nonbinary graph-based codes.
This work provides a generalization of absorbing sets for linear channel codes over non-binary alphabets. In a graphical representation of a non-binary channel code, an absorbing set can be described by a collection of topological and edge labeling conditions. In the non-binary case the equations relating neighboring variable and check nodes are over a non-binary field, and the edge weights are given by the non-zero elements of that non-binary field. As a consequence, it becomes more difficult for a given structure to satisfy the absorbing set constraints. This observation in part explains the superior performance of non-binary codes over their binary counterparts. We first show that, as the field order size increases, the ratio of trapping sets that satisfy the structural conditions of absorbing sets decreases. This suggests that a trapping set-only performance estimation of non-binary codes may not be as accurate in the error floor/high reliability regime. By using both insights from graph theory and combinatorial techniques, we establish the asymptotic distribution of non-binary elementary absorbing sets for regular code ensembles. Finally, we provide design guidelines for finite-length non-binary codes free of small absorbing sets.
In this work, we first provide the definition of absorbing sets for linear channel codes over non-binary alphabets. In a graphical representation of a non-binary channel code, an absorbing set can be described by a collection of topological and edge labeling conditions. In the non-binary case, the equations relating neighboring variable and check nodes are over a non-binary field, and the edge weights are given by the non-zero elements of that non-binary field. As a consequence, it becomes more difficult for a given structure to satisfy the absorbing set constraints compared to the binary case. This observation in part explains the superior performance of non-binary codes over their binary counterparts. We show that the conditions in the non-binary absorbing set definition can be simplified in the case of non-binary elementary absorbing sets. Based on these simplified conditions, we provide design guidelines for finite-length non-binary codes free of small non-binary elementary absorbing sets. These guidelines demonstrate that even under the preserved topology, the performance of a non-binary graph-based code in the error floor region can be substantially improved by manipulating edge weights so as to avoid small absorbing sets. Our various simulation results suggest that the proposed non-binary absorbing set definition is useful for a range of code constructions and decoders. Finally, by using both insights from graph theory and combinatorial techniques, we establish the asymptotic distribution of non-binary elementary absorbing sets for regular code ensembles.
Among the recent advances in the area of low-density parity-check (LDPC) codes, protograph-based LDPC codes have the advantages of a simple design procedure and highly structured encoders and decoders. These advantages can also be exploited in the design of protograph-based generalized LDPC (G-LDPC) codes. In this dissertation we provide analytical tools which aid the design of protograph-based LDPC and G-LDPC codes. Specifically, we propose a method for computing the codeword-weight enumerators for finite-length protograph-based G-LDPC code ensembles, and then we consider the asymptotic case when the block-length goes to infinity. These results help the designer identify good ensembles of protograph-based G-LDPC codes in the minimum distance sense (i.e., ensembles which have minimum distances grow linearly with code length). Furthermore, good code ensembles can be characterized by good stopping set, trapping set, or pseudocodeword properties, which assist in the design of G-LDPC codes with low floors. We leverage our method for computing codeword-weight enumerators to compute stopping-set, and pseudocodeword enumerators for the finite-length and the asymptotic ensembles of protograph-based G-LDPC codes. Moreover, we introduce a method for computing trapping set enumerators for finite-length (and asymptotic) protograph-based LDPC code ensembles. Trapping set enumerators for G-LDPC codes represents a more complex problem which we do not consider here. Inspired by our method for computing trapping set enumerators for protograph-based LDPC code ensembles, we developed an algorithm for estimating the trapping set enumerators for a specific LDPC code given its parity-check matrix. We used this algorithm to enumerate trapping sets for several LDPC codes from communication standards. Finally, we study coded-modulation schemes with LDPC codes and pulse position modulation (LDPC-PPM) over the free-space optical channel. We present three different decoding schemes and compare their performances. In addition, we developed a new density evolution tool for use in the design of LDPC codes with good performances over this channel.
This paper introduces a novel class of non-binary graph-based codes, built upon graph covers of non-binary protograph LDPC codes. This new ensemble is more restrictive than previously considered constructions, but in turn is much simpler to design and implement. The simplifications in design allow for the enumeration of codeword weight, trapping set size, stopping set size, and pseudo codeword weights, both as exact quantities in the finite-length regime and as corresponding growth rates in the asymptotic setting. The presented construction and the accompanying enumerations enhance the analytical toolbox of non-binary LDPC codes.
Low-density parity-check (LDPC) codes in conjunction with iterative decoding based on message-passing algorithms have been shown to achieve excellent performance over a variety of communication channels. In this dissertation, we investigate LDPC codes under two scenarios which present different challenges for providing robust communications. First, LDPC codes are analyzed and designed for time-selective, frequency-nonselective complex-fading channels where both the amplitude and the phase of the transmitted signal are altered by the channel. To combat the fading, we use a pilot symbol-assisted scheme where known pilot symbols are added to the LDPC code and used to help estimate the channel at the receiver. Using asymptotic, infinite-length analysis, we investigate several iterative, message-passing, joint-estimation-and-decoding strategies; optimal energy distribution between pilot and data symbols; and LDPC code design. Several interesting results are obtained regarding unification of analysis and code design. Next, we address the issue of achieving very low error rates with practical, finite-length LDPC codes for binary erasure channels. Finite-length LDPC codes suffer from error floors which limit the achievable error rate, and this error-floor performance is determined by stopping sets. Asymptotic analysis of weight and stopping-set enumerators, for codewords and stopping sets which grow linearly with codelength, has aided in designing codes with lower error floors but does not reflect the behavior of sublinearly-sized stopping sets, which can dominate the iterative-decoding, error-floor performance. Thus, we provide a perspective on protograph-based and standard LDPC ensemble enumerators, based on analysis of stopping sets with sublinear growth, which brings new insight into sublinear stopping-set behavior, advantages of protograph structure, and effects of precoding. We show for stopping sets that grow at most logarithmically with codelength, the enumerators follow a polynomial relationship with codelength, unlike the exponential relationship for linearly-growing stopping sets, and this polynomial relationship can be approximately captured by a single parameter. Further, we begin to address the question, Given finite stopping-set sizes and finite codelengths, do the stopping sets follow the behavior predicted by linear or sublinear analysis?