## 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

The Hamming codes | 81 |

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

BCH codes binary code binary Golay code binary Hamming code binary linear code binary symmetric channel calculate Chapter code of Example code of length coding theory columns of H construct Corollary coset leaders cyclic code d)-code defined denote digits distinct dual code encoded entry equations equivalent error vector Exercise finite field given gives Hamming code Hamming code Ham Hence irreducible polynomials k}-code Latin squares Lemma linear code linearly independent MacWilliams and Sloane matrix H maxa MDS codes minimum distance modulo MOLS of order non-zero codeword non-zero elements odd weight optimal order q orthogonal pair of MOLS parameters parity-check matrix perfect codes prime number prime power proof of Theorem q-ary received vector Remark repetition code result rows of G scalar multiple Show single error sphere-packing bound standard array standard form subspace symbols syndrome ternary Theorem 8.4 values weight enumerator zero