An optical orthogonal code (OOC) is a family of (0,1) sequences with good auto- and cross-correlation properties, i.e the autocorrelation of each sequence exhibits the `thumbtack' shape and the cross correlation between any two sequences remains low throughout. The use of optical orthogonal codes enables a large number of asynchronous users to transmit information efficiently and reliably. The thumbtack-shaped autocorrelation facilitates the detection of the desired signal, and low-profile cross correlation reduces interference from unwanted signals. Theoretical upper and lower bounds on the maximum possible size of OOCs are derived. Methods for the design and analysis of OOCs, using tools from projective geometry, the greedy algorithm, iterative constructions, algebraic coding theory, block design, and various other combinational disciplines, are discussed

Optical orthogonal codes: design, analysis and applications

Optical orthogonal codes: design, analysis, and applications

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

Numbered undirected graphs are becoming an increasingly useful family of mathematical models for a broad range of applications. They have found usage in various coding theory problems, including the design of good radar-type codes, synch-set codes and convolutional codes with optimal autocorrelation properties. They facilitate the optimal nonstandard encodings of integers. They have also been applied to determining ambiguities in X-ray crystallographic analysis, to design of a communication network addressing system, to determination of optimal circuit layouts, and to problems in additive number theory. An attempt has been made to systematically present all of these diverse applications in a unifying framework and to indicate the existence of additional applications and to suggest directions for additional research.

Applications of numbered undirected graphs

Gleason has described the general form that the weight distribution of a self-dual code over GF (2) and GF (3) can have. We give an explicit formula for this weight distribution when the minimum distance d between codewords is made as large as possible. It follows that for self-dual codes of length n over GF (2) with all weights divisible by 4, d ⩽ 4[ n /24] + 4; and for self-dual codes over GF (3), d ⩽ 3[ n /12] + 3; where the square brackets denote the integer part. These results improve on the Elias bound. A table of this extremal weight distribution is given in the binary case for n ⩽ 200 and n = 256.

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An Upper Bound for Self-Dual Codes

A class of binary recurrent codes for correcting independent errors is given which has guaranteed error-limiting properties. These codes can be simply decoded using threshold decoding, and will recover from any decoding error caused by either an uncorrectable transmission error or a temporary malfunction of the encoder or decoder. A number of such codes are given along with a synthesis procedure. The results of a computer simulation are given which indicate that these codes perform better in some situations than other codes using threshold decoding.

A. Bernstein

Papers

A class of binary recurrent codes with limited error propagation