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 Euclidean and projective geometries over finite fields. Codes of these four classes have good minimum distances and their Tanner (1981) graphs have girth 6. Finite-geometry LDPC codes can be decoded in various 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 decibel away from the Shannon theoretical limit with iterative decoding.

/pdf/low-density-parity-check-codes-based-on-finite-geometries-a-4hq963rleu.pdf

Low-density parity-check codes based on finite geometries: a rediscovery and new results

Two simplified versions of the belief propagation algorithm for fast iterative decoding of low-density parity check codes on the additive white Gaussian noise channel are proposed. Both versions are implemented with real additions only, which greatly simplifies the decoding complexity of belief propagation in which products of probabilities have to be computed. Also, these two algorithms do not require any knowledge about the channel characteristics. Both algorithms yield a good performance-complexity trade-off and can be efficiently implemented in software as well as in hardware, with possibly quantized received values.

/pdf/reduced-complexity-iterative-decoding-of-low-density-parity-1mvw7z7d8k.pdf

Reduced complexity iterative decoding of low-density parity check codes based on belief propagation

We present a class of convolutional codes defined by a low-density parity-check matrix and an iterative algorithm for decoding these codes The performance of this decoding is close to the performance of turbo decoding Our simulation shows that for the rate R=1/2 binary codes, the performance is substantially better than for ordinary convolutional codes with the same decoding complexity per information bit As an example, we constructed convolutional codes with memory M=1025, 2049, and 4097 showing that we are about 1 dB from the capacity limit at a bit-error rate (BER) of 10/sup -5/ and a decoding complexity of the same magnitude as a Viterbi decoder for codes having memory M=10

Time-varying periodic convolutional codes with low-density parity-check matrix

The goal of the present paper is the derivation of a framework for the finite-length analysis of message-passing iterative decoding of low-density parity-check codes. To this end we introduce the concept of graph-cover decoding. Whereas in maximum-likelihood decoding all codewords in a code are competing to be the best explanation of the received vector, under graph-cover decoding all codewords in all finite covers of a Tanner graph representation of the code are competing to be the best explanation. We are interested in graph-cover decoding because it is a theoretical tool that can be used to show connections between linear programming decoding and message-passing iterative decoding. Namely, on the one hand it turns out that graph-cover decoding is essentially equivalent to linear programming decoding. On the other hand, because iterative, locally operating decoding algorithms like message-passing iterative decoding cannot distinguish the underlying Tanner graph from any covering graph, graph-cover decoding can serve as a model to explain the behavior of message-passing iterative decoding. Understanding the behavior of graph-cover decoding is tantamount to understanding the so-called fundamental polytope. Therefore, we give some characterizations of this polytope and explain its relation to earlier concepts that were introduced to understand the behavior of message-passing iterative decoding for finite-length codes.

Graph-Cover Decoding and Finite-Length Analysis of Message-Passing Iterative Decoding of LDPC Codes

Preface Acknowledgements List of Abbreviations List of Symbols 1 Introduction to Digital Communications 11 Basic System Model 12 Characteristics of Mobile Radio Channels 13 Signal Detection 131 Optimal Decision Criteria 14 Digital Linear Modulation 15 Diversity 16 Summary 2 Information Theory 21 Basic Definitions 22 Channel Coding Theorem for SISO Channels 23 Channel Capacity of MIMO Systems 24 Channel Capacity for Multi-User Communications 25 Summary 3 Forward Error Correction Coding 31 Introduction 32 Linear Block Codes 33 Convolutional Codes 34 Soft-Output Decoding of Binary Codes 35 Performance Evaluation of Linear Codes 36 Concatenated Codes 37 Low Density Parity Check (LDPC) Codes 38 Summary 4 Code Division Multiple Access 41 Fundamentals 42 OFDM-CDMA 43 Low Rate Channel Coding in CDMA Systems 44 Uplink Capacity of CDMA Systems 45 Summary 5 Multi-User Detection in CDMA Systems 51 Optimum Detection 52 Linear Multi-User Detection 53 Nonlinear Iterative Multi-User Detection 54 Combining Linear MUD and Nonlinear SIC 55 Summary 6 Multiple Antenna Systems 61 Introduction  62 Spatial Diversity Concepts 63 Multi-Layer Transmission 64 Linear Dispersion Codes 65 Information Theoretic Analysis 66 Summary Appendix A: Channel Models A1 Equivalent Baseband Representation A2 Typical Propagation Profiles for Outdoor Mobile Radio Channels A3 Moment Generating Function for Ricean Fading Appendix B: Derivations for Information Theory B1 Chain Rule for Entropies B2 Chain Rule for Information B3 Data Processing Theorem Appendix C: Linear Algebra C1 Selected Basics C2 Householder Reflections and Givens Rotation C3 LLL Lattice-Reduction Bibliography Index

Wireless Communications over Mimo Channels: Applications to Cdma And Multiple Antenna Systems

Iterative decoding methods have gained interest, initiated by the results of the so-called "turbo" codes. The theoretical description of this decoding, however, seems to be difficult. Therefore, we study the iterative decoding of block codes. First, we discuss the iterative decoding algorithms developed by Gallager (1962), Battail et al. (1979), and Hagenauer et al. (1996). Based on their results, we propose a decoding algorithm which only uses parity check vectors of minimum weight. We give the relation of this iterative decoding to one-step majority-logic decoding, and interpret it as gradient optimization. It is shown that the used parity check set defines the region where the iterative decoding decides on a particular codeword. We make plausible that, in almost all cases, the iterative decoding converges to a codeword after some iterations. We derive a computationally efficient implementation using the minimal trellis representing the used parity check set. Simulations illustrate that our algorithm gives results close to soft decision maximum likelihood (SDML) decoding for many code classes like BCH codes. Reed-Muller codes, quadratic residue codes, double circulant codes, and cyclic finite geometry codes. We also present simulation results for product codes and parallel concatenated codes based on block codes.

R. Lucas

Papers

On iterative soft-decision decoding of linear binary block codes and product codes