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

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

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

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