Deriving Good LDPC Convolutional Codes from LDPC Block Codes

Low-density parity-check (LDPC) convolutional codes are capable of achieving excellent performance with low encoding and decoding complexity. In this paper we discuss several graph-cover-based methods for deriving families of time-invariant and time-varying LDPC convolutional codes from LDPC block codes and show how earlier proposed LDPC convolutional code constructions can be presented within this framework. Some of the constructed convolutional codes significantly outperform the underlying LDPC block codes. We investigate some possible reasons for this "convolutional gain," and we also discuss the --- mostly moderate --- decoder cost increase that is incurred by going from LDPC block to LDPC convolutional codes.Comment: Submitted to IEEE Transactions on Information Theory, April 2010; revised August 2010, revised November 2010 (essentially final version). (Besides many small changes, the first and second revised versions contain corrected entries in Tables I and II.

Codes on Graphs and More

Modern communication systems strive to achieve reliable and efficient information transmission and storage with affordable complexity. Hence, efficient low-complexity channel codes providing low probabilities for erroneous receptions are needed. Interpreting codes as graphs and graphs as codes opens new perspectives for constructing such channel codes. Low-density parity-check (LDPC) codes are one of the most recent examples of codes defined on graphs, providing a better bit error probability than other block codes, given the same decoding complexity. After an introduction to coding theory, different graphical representations for channel codes are reviewed. Based on ideas from graph theory, new algorithms are introduced to iteratively search for LDPC block codes with large girth and to determine their minimum distance. In particular, new LDPC block codes of different rates and with girth up to 24 are presented. Woven convolutional codes are introduced as a generalization of graph-based codes and an asymptotic bound on their free distance, namely, the Costello lower bound, is proven. Moreover, promising examples of woven convolutional codes are given, including a rate 5/20 code with overall constraint length 67 and free distance 120. The remaining part of this dissertation focuses on basic properties of convolutional codes. First, a recurrent equation to determine a closed form expression of the exact decoding bit error probability for convolutional codes is presented. The obtained closed form expression is evaluated for various realizations of encoders, including rate 1/2 and 2/3 encoders, of as many as 16 states. Moreover, MacWilliams-type identities are revisited and a recursion for sequences of spectra of truncated as well as tailbitten convolutional codes and their duals is derived. Finally, the dissertation is concluded with exhaustive searches for convolutional codes of various rates with either optimum free distance or optimum distance profile, extending previously published results

On the Minimum Distance of Generalized Spatially Coupled LDPC Codes

Families of generalized spatially-coupled low-density parity-check (GSC-LDPC) code ensembles can be formed by terminating protograph-based generalized LDPC convolutional (GLDPCC) codes. It has previously been shown that ensembles of GSC-LDPC codes constructed from a protograph have better iterative decoding thresholds than their block code counterparts, and that, for large termination lengths, their thresholds coincide with the maximum a-posteriori (MAP) decoding threshold of the underlying generalized LDPC block code ensemble. Here we show that, in addition to their excellent iterative decoding thresholds, ensembles of GSC-LDPC codes are asymptotically good and have large minimum distance growth rates.Comment: Submitted to the IEEE International Symposium on Information Theory 201

A Simple Proof of Maxwell Saturation for Coupled Scalar Recursions

Low-density parity-check (LDPC) convolutional codes (or spatially-coupled codes) were recently shown to approach capacity on the binary erasure channel (BEC) and binary-input memoryless symmetric channels. The mechanism behind this spectacular performance is now called threshold saturation via spatial coupling. This new phenomenon is characterized by the belief-propagation threshold of the spatially-coupled ensemble increasing to an intrinsic noise threshold defined by the uncoupled system. In this paper, we present a simple proof of threshold saturation that applies to a wide class of coupled scalar recursions. Our approach is based on constructing potential functions for both the coupled and uncoupled recursions. Our results actually show that the fixed point of the coupled recursion is essentially determined by the minimum of the uncoupled potential function and we refer to this phenomenon as Maxwell saturation. A variety of examples are considered including the density-evolution equations for: irregular LDPC codes on the BEC, irregular low-density generator matrix codes on the BEC, a class of generalized LDPC codes with BCH component codes, the joint iterative decoding of LDPC codes on intersymbol-interference channels with erasure noise, and the compressed sensing of random vectors with i.i.d. components.Comment: This article is an extended journal version of arXiv:1204.5703 and has now been accepted to the IEEE Transactions on Information Theory. This version adds additional explanation for some details and also corrects a number of small typo

Error-correction on non-standard communication channels

Many communication systems are poorly modelled by the standard channels assumed in the information theory literature, such as the binary symmetric channel or the additive white Gaussian noise channel. Real systems suffer from additional problems including time-varying noise, cross-talk, synchronization errors and latency constraints. In this thesis, low-density parity-check codes and codes related to them are applied to non-standard channels. First, we look at time-varying noise modelled by a Markov channel. A low-density parity-check code decoder is modified to give an improvement of over 1dB. Secondly, novel codes based on low-density parity-check codes are introduced which produce transmissions with Pr(bit = 1) ≠ Pr(bit = 0). These non-linear codes are shown to be good candidates for multi-user channels with crosstalk, such as optical channels. Thirdly, a channel with synchronization errors is modelled by random uncorrelated insertion or deletion events at unknown positions. Marker codes formed from low-density parity-check codewords with regular markers inserted within them are studied. It is shown that a marker code with iterative decoding has performance close to the bounds on the channel capacity, significantly outperforming other known codes. Finally, coding for a system with latency constraints is studied. For example, if a telemetry system involves a slow channel some error correction is often needed quickly whilst the code should be able to correct remaining errors later. A new code is formed from the intersection of a convolutional code with a high rate low-density parity-check code. The convolutional code has good early decoding performance and the high rate low-density parity-check code efficiently cleans up remaining errors after receiving the entire block. Simulations of the block code show a gain of 1.5dB over a standard NASA code

Searching for Voltage Graph-Based LDPC Tailbiting Codes with Large Girth

The relation between parity-check matrices of quasi-cyclic (QC) low-density parity-check (LDPC) codes and biadjacency matrices of bipartite graphs supports searching for powerful LDPC block codes. Using the principle of tailbiting, compact representations of bipartite graphs based on convolutional codes can be found. Bounds on the girth and the minimum distance of LDPC block codes constructed in such a way are discussed. Algorithms for searching iteratively for LDPC block codes with large girth and for determining their minimum distance are presented. Constructions based on all-ones matrices, Steiner Triple Systems, and QC block codes are introduced. Finally, new QC regular LDPC block codes with girth up to 24 are given.Comment: Submitted to IEEE Transactions on Information Theory in February 201

Quasi-Cyclic Asymptotically Regular LDPC Codes

Families of "asymptotically regular" LDPC block code ensembles can be formed by terminating (J,K)-regular protograph-based LDPC convolutional codes. By varying the termination length, we obtain a large selection of LDPC block code ensembles with varying code rates, minimum distance that grows linearly with block length, and capacity approaching iterative decoding thresholds, despite the fact that the terminated ensembles are almost regular. In this paper, we investigate the properties of the quasi-cyclic (QC) members of such an ensemble. We show that an upper bound on the minimum Hamming distance of members of the QC sub-ensemble can be improved by careful choice of the component protographs used in the code construction. Further, we show that the upper bound on the minimum distance can be improved by using arrays of circulants in a graph cover of the protograph.Comment: To be presented at the 2010 IEEE Information Theory Workshop, Dublin, Irelan

Design and Performance of Rate-compatible Non-Binary LDPC Convolutional Codes

In this paper, we present a construction method of non-binary low-density parity-check (LDPC) convolutional codes. Our construction method is an extension of Felstroem and Zigangirov construction for non-binary LDPC convolutional codes. The rate-compatibility of the non-binary convolutional code is also discussed. The proposed rate-compatible code is designed from one single mother (2,4)-regular non-binary LDPC convolutional code of rate 1/2. Higher-rate codes are produced by puncturing the mother code and lower-rate codes are produced by multiplicatively repeating the mother code. Simulation results show that non-binary LDPC convolutional codes of rate 1/2 outperform state-of-the-art binary LDPC convolutional codes with comparable constraint bit length. Also the derived low-rate and high-rate non-binary LDPC convolutional codes exhibit good decoding performance without loss of large gap to the Shannon limits.Comment: 8 pages, submitted to IEICE transactio