We discuss three structures of modified low-density parity-check (LDPC) code ensembles designed for transmission over arbitrary discrete memoryless channels. The first structure is based on the well known binary LDPC codes following constructions proposed by Gallager and McEliece, the second is based on LDPC codes of arbitrary (q-ary) alphabets employing modulo-q addition, as presented by Gallager, and the third is based on LDPC codes defined over the field GF(q). All structures are obtained by applying a quantization mapping on a coset LDPC ensemble. We present tools for the analysis of non-binary codes and show that all configurations, under maximum-likelihood decoding, are capable of reliable communication at rates arbitrarily close to channel capacity of any discrete memoryless channel. We discuss practical iterative decoding of our structures and present simulation results for the AWGN channel confirming the effectiveness of the codes.

We derive both upper and lower bounds on the decoding error probability of ML decoded LDPC codes. The results hold for any binary-input symmetric-output channel. Our results in-dicate that for various appropriately chosen ensembles of LDPC codes, reliable communication is possible up to channel capacity. However, the ensemble averaged decoding error probability decreases polynomially, and not exponentially. The lower and upper bounds coincide asymp-totically, thus showing the tightness of the bounds. However, for ensembles with suitably chosen parameters, the error probability of almost all codes is exponentially decreasing, with an error exponent that can be set arbitrarily close to the standard random coding exponent. Index Terms- Code ensembles, Error exponent, Low density parity check (LDPC) codes. I

Abstract—This letter explains the effect of graph connectivity on error-floor performance of low-density parity-check (LDPC) codes under message-passing decoding. A new metric, called extrinsic message degree (EMD), measures cycle connectivity in bipartite graphs of LDPC codes. Using an easily computed estimate of EMD, we propose a Viterbi-like algorithm that selectively avoids small cycle clusters that are isolated from the rest of the graph. This algorithm is different from conventional girth conditioning by emphasizing the connectivity as well as the length of cycles. The algorithm yields codes with error floors that are orders of magnitude below those of random codes with very small degradation in capacity-approaching capability. Index Terms—Error floor, extrinsic message degree (EMD), graph cycles, irregular low-density parity-check (LDPC) codes, iterative decoding, message passing, stopping sets, unstructured graph construction. I.

This paper presents a geometric approach to the construction of low density parity check (LDPC) codes. Four classes of LDPC codes are constructed based on the lines and points of Eu-clidean and projective geometries over finite fields. Codes of these four classes have good minimum distances and their Tanner graphs have girth 6. Finite geometry LDPC codes can be decoded in var-ious ways, ranging from low to high decoding complexity and from reasonably good to very good performance. They perform very well with iterative decoding. Furthermore, they can be put in either cyclic or quasi-cyclic form. Consequently, their encoding can be achieved in linear time and implemented with simple feedback shift registers. This advantage is not shared by other LDPC codes in general and is important in practice. Finite geometry LDPC codes can be extended and shortened in various ways to obtain other good LDPC codes. Several techniques of extension and shortening are presented. Long extended finite geometry LDPC codes have been constructed and they achieve a performance only a few tenths of a dB away from the Shannon theoretical limit with iterative decoding.

A new method for analyzing low density parity check (LDPC) codes and low density generator matrix (LDGM) codes under bit maximum a posteriori probability (MAP) decoding is introduced. The method is based on a rigorous approach to spin glasses developed by Francesco Guerra. It allows to construct lower bounds on the entropy of the transmitted message conditional to the received one. Based on heuristic statistical mechanics calculations, we conjecture such bounds to be tight. The result holds for standard irregular ensembles when used over binary input output symmetric channels. The method is first developed for Tanner graph ensembles with Poisson left degree distribution. It is then generalized to ‘multi-Poisson ’ graphs, and, by a completion procedure, to arbitrary degree distribution.

In this paper, we review recent developments concerning the application of lowdensity parity-check (LDPC) codes to the Gilbert-Elliott (GE) channel. Firstly, we discuss the analysis of LDPC estimation-decoding in these channels using density evolution. We show that the required conditions of density evolution are satisfied in the GE channel, and that analysis demonstrates that large potential gains over the memoryless assumption. We also give results which mitigate the complexity of characterizing the GE parameter space using DE. Subsequently, we give a design tool for finding good degree sequences for irregular LDPC codes in the GE channel.

We give a linear programming (LP) decoder that achieves the capacity (optimal rate) of a wide range of probabilistic binary communication channels. This is the first such result for LP decoding. More generally, as far as the authors are aware this is the first known polynomial-time capacity-achieving decoder with the maximum-likelihood (ML) certificate property---where output codewords come with a proof of optimality.

We derive upper bounds on the rate of low density parity check (LDPC) codes for which reliable communication is achievable. We rst generalize Gallager's bound to a general binaryinput symmetric-output channel. We then proceed to derive tighter bounds. We also derive upper bounds on the rate as a function of the minimum distance of the code. We consider both individual codes and ensembles of codes. Index Terms - Low density parity check (LDPC) codes, iterative decoding, maximum-likelihood decoding, error probability, minimum distance. I

We consider Gallager’s soft decoding (belief propagation) algorithm for decoding low den-sity parity check (LDPC) codes, when applied to an arbitrary binary-input symmetric-output channel. By considering the expected values of the messages, we derive both lower and upper bounds on the performance of the algorithm. We also derive various properties of the decoding algorithm, such as a certain robustness to the details of the channel noise. Our results apply both to regular and irregular LDPC codes. Index Terms- Belief propagation, Iterative decoding, Low density parity check (LDPC) codes, Sum product algorithm. I

We present sparse-graph codes appropriate for use in quantum error-correction. Quantum error-correcting codes based on sparse graphs are of interest for three reasons. First, the best codes currently known for classical channels are based on sparse graphs. Second, sparsegraph codes keep the number of quantum interactions associated with the quantum error-correction process small: a constant number per quantum bit, independent of the blocklength. Third, sparse-graph codes often offer great flexibility with respect to blocklength and rate. We believe some of the codes we present are unsurpassed by previously published quantum error-correcting codes.