## Error-correcting codes and finite fieldsThis book provides the reader with all the tools necessary to implement modern error-processing techniques. It assumes only a basic knowledge of linear algebra and develops the mathematical theory in parallel with the codes. Central to the text are worked examples which motivate and explain the theory. The book is in four parts. The first introduces the basic ideas of coding theory. The second and third parts cover the theory of finite fields and give a detailed treatment of BCH and Reed-Solomon codes. These parts are linked by their use of Euclid's algorithm as a central technique. The fourth part is devoted to Goppa codes, both classical and geometric, concluding with the Skorobogatov-Vladut error processor. A special feature of this part is a simplified (but rigorous) treatment of the geometry of curves. The book is intended for the advanced instruction of engineers and computer scientists. |

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### Contents

Introduction | 3 |

Linear codes | 27 |

Definition of linear codes and fields Dimension and rate The generator | 40 |

Copyright | |

24 other sections not shown

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### Common terms and phrases

BCH codes BCH(fc binary code block length calculate chapter check matrix code polynomial code word coefficients columns construct Corollary correct coset decoding defined degree less denote dimension divides divisor encoding entries error evaluator error locator error patterns error processor error word error-correcting error-processing errors occurred errors of weight Euclid's algorithm Euclidean domain Example Exercise field F field of order finite field follows formula fundamental equation GC(P geometric Goppa codes Golay code Goppa codes Hamming codes Hence highest common factor integers irreducible polynomials Klein quartic Lemma Let F linear code linearly independent logarithms message word minimal polynomial modulo non-zero elements order q points polynomial f(x polynomials of degree primitive element produce Proposition Let prove rational functions received word Reed—Solomon codes remainder roots row leaders single errors standard array subfield Suppose syndrome polynomial Theorem Let transmitted triple check code unique vector space zero