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.

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

It is well known that if the data rate is chosen below the available channel capacity, error-free communication is possible. Furthermore, numerous practical error-correction coding techniques exist which can be chosen to meet the user's reliability constraints. However, a basic problem in designing a reliable digital communication system is still the choice of the actual code rate. While the popular rate-1/2 code rate is a reasonable, but not optimum, choice for additive Gaussian noise channels, its selection is far from optimum for channels where a high percentage of the transmitted bits are destroyed by interference. Code combining represents a technique of matching the code rate to the prevailing channel conditions. Information is transmitted in packet formats which are encoded with a relatively high-rate code, e.g., rate 1/2, which can be repeated to Obtain reliable communications when the redundancy in a rate-1/2 code is not sufficient to overcome the channel interference. The receiver combines noisy packets (code combining) to obtain a packet with a code rate which is low enough such that reliable communication is possible even for channels with extremely high error rates. By combining the minimum number of packets needed to overcome the channel conditions, the receiver optimizes the code rate and minimizes the delay required to decode a given packet. Thus, the receiver adapts to the actual jammer-to-signal (J/S) ratio which is critical when the level of interference J is not known a priori.

Code Combining--A Maximum-Likelihood Decoding Approach for Combining an Arbitrary Number of Noisy Packets

ERROR DETECTION incorporated with automatic-repeatrequest (ARQ) is widely used for error control in data communications systems. This method of error control is simple and provides high system reliability. If a properly chosen code is used for error detection, virtually error-free data transmission can be attained. This paper surveys various types of ARQ and hybrid ARQ schemes, and error detection using linear block codes.

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Automatic-repeat-request error-control schemes

Automatic-repeat-request error control schemes

The need to transmit and store massive amounts of data reliably and without error is a vital part of modern communications systems. Error-correcting codes play a fundamental role in minimising data corruption caused by defects such as noise, interference, crosstalk and packet loss. This book provides an accessible introduction to the basic elements of algebraic codes, and discusses their use in a variety of applications. The author describes a range of important coding techniques, including Reed-Solomon codes, BCH codes, trellis codes, and turbocodes. Throughout the book, mathematical theory is illustrated by reference to many practical examples. The book was first published in 2003 and is aimed at graduate students of electrical and computer engineering, and at practising engineers whose work involves communications or signal processing.

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Algebraic Codes for Data Transmission

A new procedure for handling retransmissions in a selective-repeat ARQ system is proposed. This procedure can operate with a receive buffer of minimal size; in addition it places little computational load on the transmit and receive processors. The procedure is simple enough that its throughput can be calculated exactly. Analysis of this strategy shows that: 1)it yields higher throughput than earlier ARQ techniques; 2) for modest receive buffer size, its throughput differs little from channel capacity; 3) as buffer size increases, throughput approaches channel capacity. The final section of the paper considers the performance of ARQ systems on channels in which errors occur in bursts. It indicates that on reasonably good channels, error burstiness has little effect on throughput.

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An Improved Selective-Repeat ARQ Strategy

In this paper a class of nonprimitive cyclic codes quite similar in structure to the original Reed-Muller codes is presented. These codes, referred to herein as nonprimitive Reed-Muller codes, are shown to possess many of the properties of the primitive codes. Specifically, two major results are presented. First the code length, number of information symbols, and minimum distance are shown to be related by means of a parameter known as the order of the code. These relationships show that for given values of code length and rate the codes have relatively large minimum distances. It is also shown that the codes are subcodes of the BCH codes of the same length and guaranteed minimum distance; thus in general the codes are not as powerful as the BCH codes. However, for most interesting values of code length and rate the difference between the two types of codes is slight. The second result is the observation that the codes can be decoded with a variation of the original algorithm proposed by Reed for the Reed-Muller codes. In other words, they are L -step orthogonalizable. Because of their large minimum distances and the simplicity of their decoders, nonprimitive Reed-Muller codes seem attractive for use in error-control systems requiring multiple random-error correction.

New generalizations of the Reed-Muller codes--II: Nonprimitive codes

This paper presents an algebraic technique for decoding binary block codes in situations where the demodulator quantizes the received signal space into Q > 2 regions. The method, referred to as weighted erasure decoding (WED), is applicable in principle to any block code for which a binary decoding procedure is known.

Decoding binary block codes on Q -ary output channels

Correction to "New Generalizations of the Reed-Muller Codes - Part II: Nonprimitive Codes"

Justesen has shown that concatenating a class of binary codes with a Reed-Solomon (RS) code produces asymptotically good codes. For low rates, the value of the ratio of minimum distance to code length (d/n) for such codes is substantially lower than that known to be achievable by the Zyablov bound. In this paper, we present a small class of binary codes with some useful properties. This class is then used in Justesen's construction to produce codes that have relatively large values of d/n for low rates.

Jr. E. Weldon

Papers

An Improved Selective-Repeat ARQ Strategy

New generalizations of the Reed-Muller codes--II: Nonprimitive codes

Decoding binary block codes on Q -ary output channels

Correction to "New Generalizations of the Reed-Muller Codes - Part II: Nonprimitive Codes"

Some results on the problem of constructing asymptotically good error-correcting codes