Applications of correlation inequalities to low density graphical codes

This contribution is based on the contents of a talk delivered at the Next-SigmaPhi conference held in Crete in August 2005. It is adressed to an audience of physicists with diverse horizons and does not assume any background in communications theory. Capacity approaching error correcting codes for channel communication known as Low Density Parity Check (LDPC) codes have attracted considerable attention from coding theorists in the last decade. Surprisingly strong connections with the theory of diluted spin glasses have been discovered. In this work we elucidate one new connection, namely that a class of correlation inequalities valid for gaussian spin glasses can be applied to the theoretical analysis of LDPC codes. This allows for a rigorous comparison between the so called (optimal) maximum a posteriori and the computationaly efficient belief propagation decoders. The main ideas of the proofs are explained and we refer to recent works for the more lengthy technical details.Comment: 11 pages, 3 figure

Density Evolution for Asymmetric Memoryless Channels

Density evolution is one of the most powerful analytical tools for low-density parity-check (LDPC) codes and graph codes with message passing decoding algorithms. With channel symmetry as one of its fundamental assumptions, density evolution (DE) has been widely and successfully applied to different channels, including binary erasure channels, binary symmetric channels, binary additive white Gaussian noise channels, etc. This paper generalizes density evolution for non-symmetric memoryless channels, which in turn broadens the applications to general memoryless channels, e.g. z-channels, composite white Gaussian noise channels, etc. The central theorem underpinning this generalization is the convergence to perfect projection for any fixed size supporting tree. A new iterative formula of the same complexity is then presented and the necessary theorems for the performance concentration theorems are developed. Several properties of the new density evolution method are explored, including stability results for general asymmetric memoryless channels. Simulations, code optimizations, and possible new applications suggested by this new density evolution method are also provided. This result is also used to prove the typicality of linear LDPC codes among the coset code ensemble when the minimum check node degree is sufficiently large. It is shown that the convergence to perfect projection is essential to the belief propagation algorithm even when only symmetric channels are considered. Hence the proof of the convergence to perfect projection serves also as a completion of the theory of classical density evolution for symmetric memoryless channels.Comment: To appear in the IEEE Transactions on Information Theor

Windowed Decoding of Protograph-based LDPC Convolutional Codes over Erasure Channels

We consider a windowed decoding scheme for LDPC convolutional codes that is based on the belief-propagation (BP) algorithm. We discuss the advantages of this decoding scheme and identify certain characteristics of LDPC convolutional code ensembles that exhibit good performance with the windowed decoder. We will consider the performance of these ensembles and codes over erasure channels with and without memory. We show that the structure of LDPC convolutional code ensembles is suitable to obtain performance close to the theoretical limits over the memoryless erasure channel, both for the BP decoder and windowed decoding. However, the same structure imposes limitations on the performance over erasure channels with memory.Comment: 18 pages, 9 figures, accepted for publication in the IEEE Transactions on Information Theor

The Generalized Area Theorem and Some of its Consequences

There is a fundamental relationship between belief propagation and maximum a posteriori decoding. The case of transmission over the binary erasure channel was investigated in detail in a companion paper. This paper investigates the extension to general memoryless channels (paying special attention to the binary case). An area theorem for transmission over general memoryless channels is introduced and some of its many consequences are discussed. We show that this area theorem gives rise to an upper-bound on the maximum a posteriori threshold for sparse graph codes. In situations where this bound is tight, the extrinsic soft bit estimates delivered by the belief propagation decoder coincide with the correct a posteriori probabilities above the maximum a posteriori threshold. More generally, it is conjectured that the fundamental relationship between the maximum a posteriori and the belief propagation decoder which was observed for transmission over the binary erasure channel carries over to the general case. We finally demonstrate that in order for the design rate of an ensemble to approach the capacity under belief propagation decoding the component codes have to be perfectly matched, a statement which is well known for the special case of transmission over the binary erasure channel.Comment: 27 pages, 46 ps figure

Modern Coding Theory: The Statistical Mechanics and Computer Science Point of View

These are the notes for a set of lectures delivered by the two authors at the Les Houches Summer School on `Complex Systems' in July 2006. They provide an introduction to the basic concepts in modern (probabilistic) coding theory, highlighting connections with statistical mechanics. We also stress common concepts with other disciplines dealing with similar problems that can be generically referred to as `large graphical models'. While most of the lectures are devoted to the classical channel coding problem over simple memoryless channels, we present a discussion of more complex channel models. We conclude with an overview of the main open challenges in the field.Comment: Lectures at Les Houches Summer School on `Complex Systems', July 2006, 44 pages, 25 ps figure

Distance Properties of Short LDPC Codes and their Impact on the BP, ML and Near-ML Decoding Performance

Parameters of LDPC codes, such as minimum distance, stopping distance, stopping redundancy, girth of the Tanner graph, and their influence on the frame error rate performance of the BP, ML and near-ML decoding over a BEC and an AWGN channel are studied. Both random and structured LDPC codes are considered. In particular, the BP decoding is applied to the code parity-check matrices with an increasing number of redundant rows, and the convergence of the performance to that of the ML decoding is analyzed. A comparison of the simulated BP, ML, and near-ML performance with the improved theoretical bounds on the error probability based on the exact weight spectrum coefficients and the exact stopping size spectrum coefficients is presented. It is observed that decoding performance very close to the ML decoding performance can be achieved with a relatively small number of redundant rows for some codes, for both the BEC and the AWGN channels