Design of Non-Binary Quasi-Cyclic LDPC Codes by ACE Optimization

An algorithm for constructing Tanner graphs of non-binary irregular quasi-cyclic LDPC codes is introduced. It employs a new method for selection of edge labels allowing control over the code's non-binary ACE spectrum and resulting in low error-floor. The efficiency of the algorithm is demonstrated by generating good codes of short to moderate length over small fields, outperforming codes generated by the known methods.Comment: Accepted to 2013 IEEE Information Theory Worksho

A mathematical tool for constructing parametrizable spatially-coupled LDPC codes with cyclic structure and large girth

Spatially-coupled low-density parity-check codes (SC-LDPC) have been shown to be superior in performance than LDPC block codes for both communication and storage systems. Several heuristic construction methods for these codes have been proposed in the literature, but they allow the construction of SC-LDPC codes for only specific nodedegrees, short code length and lead to encoders/decoders with non-parametrizable complex architectures. In this work we construct a mathematical tool for generating SC-LDPC codes with arbitrary node-degrees, girth of at least six and a parity-matrix with cyclic structure. The generated codes satisfy some minimum communication performance requirements which can be previously determined and can they can also be encoded/decoded with reduced-complexity parametrizable hardware architectures. An encoder architecture with reduced memory size and reduced-complexity, known as partial-syndrome based encoder, was implemented in software and the code encodability was verified. The partial-syndrome encoder structure proposed in the literature has constrained code rate and a modified SC-LDPC code was implemented, allowing the generated codes to be encoded with the partial-syndrome encoder architecture for arbitrary rates. A reduced-complexity decoder known as window decoder was implemented in software and the code decodability was also verified.Códigos Spatially-coupled low-density parity-check (SC-LDPC) têm apresentado melhor performance do que LDPC block codes, tanto em sistemas de comunicação quanto de armazenamento. Diversos métodos heurísticos de construção para estes códigos têm sido propostos na literatura, os quais possibilitam a obtenção de códigos SC-LDPC com específicos node-degrees, pequenos comprimentos de código e necessitam codificadores/decodificadores de arquitetura complexa não-parametrizável. Neste trabalho, construiu-se uma ferramenta matemática para a geração de códigos SC-LDPC com node-degrees arbitrários, girth de no mínimo seis e matriz de paridade com estrutura cíclica. Os códigos gerados satisfazem requisitos mínimos de performance de comunicação que podem ser previamente estabelecidos e podem ser codificados/decodificados por arquiteturas de hardware parametrizáveis de complexidade reduzida. Implementou-se em software um codificador de arquitetura parametrizável com tamanho de memória reduzido e baixa complexidade, conhecido como codificador baseado em partial syndrome, e verificou-se a codificação dos códigos construídos. As arquiteturas para codificadores do tipo partial-syndrome encontradas na literatura possuem taxas de codificação não arbitrárias e por isso, modificou-se os códigos SC-LDPC construídos, permitindo que os códigos gerados possam ser codificados com o mesmo codificador do tipo partial-syndrome para taxas de codificação arbitrárias. Implementou-se em software um decodificador de complexidade reduzida, conhecido como window decoder, e verificou-se a convergência dos códigos SC-LDPC construídos

Hierarchical and High-Girth QC LDPC Codes

We present a general approach to designing capacity-approaching high-girth low-density parity-check (LDPC) codes that are friendly to hardware implementation. Our methodology starts by defining a new class of "hierarchical" quasi-cyclic (HQC) LDPC codes that generalizes the structure of quasi-cyclic (QC) LDPC codes. Whereas the parity check matrices of QC LDPC codes are composed of circulant sub-matrices, those of HQC LDPC codes are composed of a hierarchy of circulant sub-matrices that are in turn constructed from circulant sub-matrices, and so on, through some number of levels. We show how to map any class of codes defined using a protograph into a family of HQC LDPC codes. Next, we present a girth-maximizing algorithm that optimizes the degrees of freedom within the family of codes to yield a high-girth HQC LDPC code. Finally, we discuss how certain characteristics of a code protograph will lead to inevitable short cycles, and show that these short cycles can be eliminated using a "squashing" procedure that results in a high-girth QC LDPC code, although not a hierarchical one. We illustrate our approach with designed examples of girth-10 QC LDPC codes obtained from protographs of one-sided spatially-coupled codes.Comment: Submitted to IEEE Transactions on Information THeor

Algebraic Design and Implementation of Protograph Codes using Non-Commuting Permutation Matrices

Random lifts of graphs, or equivalently, random permutation matrices, have been used to construct good families of codes known as protograph codes. An algebraic analog of this approach was recently presented using voltage graphs, and it was shown that many existing algebraic constructions of graph-based codes that use commuting permutation matrices may be seen as special cases of voltage graph codes. Voltage graphs are graphs that have an element of a finite group assigned to each edge, and the assignment determines a specific lift of the graph. In this paper we discuss how assignments of permutation group elements to the edges of a base graph affect the properties of the lifted graph and corresponding codes, and present a construction method of LDPC code ensembles based on noncommuting permutation matrices. We also show encoder and decoder implementations for these codes

Construction of Near-Optimum Burst Erasure Correcting Low-Density Parity-Check Codes

In this paper, a simple, general-purpose and effective tool for the design of low-density parity-check (LDPC) codes for iterative correction of bursts of erasures is presented. The design method consists in starting from the parity-check matrix of an LDPC code and developing an optimized parity-check matrix, with the same performance on the memory-less erasure channel, and suitable also for the iterative correction of single bursts of erasures. The parity-check matrix optimization is performed by an algorithm called pivot searching and swapping (PSS) algorithm, which executes permutations of carefully chosen columns of the parity-check matrix, after a local analysis of particular variable nodes called stopping set pivots. This algorithm can be in principle applied to any LDPC code. If the input parity-check matrix is designed for achieving good performance on the memory-less erasure channel, then the code obtained after the application of the PSS algorithm provides good joint correction of independent erasures and single erasure bursts. Numerical results are provided in order to show the effectiveness of the PSS algorithm when applied to different categories of LDPC codes.Comment: 15 pages, 4 figures. IEEE Trans. on Communications, accepted (submitted in Feb. 2007

Spatially Coupled LDPC Codes Constructed from Protographs

In this paper, we construct protograph-based spatially coupled low-density parity-check (SC-LDPC) codes by coupling together a series of L disjoint, or uncoupled, LDPC code Tanner graphs into a single coupled chain. By varying L, we obtain a flexible family of code ensembles with varying rates and frame lengths that can share the same encoding and decoding architecture for arbitrary L. We demonstrate that the resulting codes combine the best features of optimized irregular and regular codes in one design: capacity approaching iterative belief propagation (BP) decoding thresholds and linear growth of minimum distance with block length. In particular, we show that, for sufficiently large L, the BP thresholds on both the binary erasure channel (BEC) and the binary-input additive white Gaussian noise channel (AWGNC) saturate to a particular value significantly better than the BP decoding threshold and numerically indistinguishable from the optimal maximum a-posteriori (MAP) decoding threshold of the uncoupled LDPC code. When all variable nodes in the coupled chain have degree greater than two, asymptotically the error probability converges at least doubly exponentially with decoding iterations and we obtain sequences of asymptotically good LDPC codes with fast convergence rates and BP thresholds close to the Shannon limit. Further, the gap to capacity decreases as the density of the graph increases, opening up a new way to construct capacity achieving codes on memoryless binary-input symmetric-output (MBS) channels with low-complexity BP decoding.Comment: Submitted to the IEEE Transactions on Information Theor

Design and Analysis of Graph-based Codes Using Algebraic Lifts and Decoding Networks

Error-correcting codes seek to address the problem of transmitting information efficiently and reliably across noisy channels. Among the most competitive codes developed in the last 70 years are low-density parity-check (LDPC) codes, a class of codes whose structure may be represented by sparse bipartite graphs. In addition to having the potential to be capacity-approaching, LDPC codes offer the significant practical advantage of low-complexity graph-based decoding algorithms. Graphical substructures called trapping sets, absorbing sets, and stopping sets characterize failure of these algorithms at high signal-to-noise ratios. This dissertation focuses on code design for and analysis of iterative graph-based message-passing decoders. The main contributions of this work include the following: the unification of spatially-coupled LDPC (SC-LDPC) code constructions under a single algebraic graph lift framework and the analysis of SC-LDPC code construction techniques from the perspective of removing harmful trapping and absorbing sets; analysis of the stopping and absorbing set parameters of hypergraph codes and finite geometry LDPC (FG-LDPC) codes; the introduction of multidimensional decoding networks that encode the behavior of hard-decision message-passing decoders; and the presentation of a novel Iteration Search Algorithm, a list decoder designed to improve the performance of hard-decision decoders. Adviser: Christine A. Kelle