## A First Course in Coding TheoryAlgebraic coding theory is a new and rapidly developing subject, popular for its many practical applications and for its fascinatingly rich mathematical structure. This book provides an elementary yet rigorous introduction to the theory of error-correcting codes. Based on courses given by the author over several years to advanced undergraduates and first-year graduated students, this guide includes a large number of exercises, all with solutions, making the book highly suitable for individual study. |

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### Contents

Codes and Latin squares | 113 |

A doubleerror correcting decimal code and | 125 |

Cyclic codes | 141 |

Weight enumerators | 165 |

The main linear coding theory problem | 175 |

MDS codes | 191 |

Concluding remarks related topics and further | 201 |

Solutions to exercises | 211 |

243 | |

249 | |

### Common terms and phrases

algorithm apply array assume binary block bound calculate called channel Chapter code of length codeword columns columns of H consider construct contains Corollary correct coset leaders cyclic code decoding defined Definition denote detect determinant digits distinct elements entry equal equations equivalent error exactly Example Exercise exists factor field finite GF(q given gives Golay code Hamming code Hence implies integer k]-code known Latin squares least Lemma linear code linearly independent MacWilliams method minimum distance MOLS of order multiplication non-zero Note obtained occurred optimal parameters parity-check matrix plane points polynomial position possible precisely probability problem Proof proved q-ary received vector Remark result rows satisfies scalar Show single space standard form Suppose symbols syndrome ternary Theorem unique values weight write zero