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

Cyclic Quantum Error-Correcting Codes and Quantum Shift Registers

We transfer the concept of linear feed-back shift registers to quantum circuits. It is shown how to use these quantum linear shift registers for encoding and decoding cyclic quantum error-correcting codes.Comment: 18 pages, 15 figures, submitted to Proc. R. Soc.

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