## Coding Theory: A First CourseCoding theory is concerned with successfully transmitting data through a noisy channel and correcting errors in corrupted messages. It is of central importance for many applications in computer science or engineering. This book gives a comprehensive introduction to coding theory whilst only assuming basic linear algebra. It contains a detailed and rigorous introduction to the theory of block codes and moves on to more advanced topics like BCH codes, Goppa codes and Sudan's algorithm for list decoding. The issues of bounds and decoding, essential to the design of good codes, features prominently. The authors of this book have, for several years, successfully taught a course on coding theory to students at the National University of Singapore. This book is based on their experiences and provides a thoroughly modern introduction to the subject. There are numerous examples and exercises, some of which introduce students to novel or more advanced material. |

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### Contents

II | 1 |

III | 5 |

V | 8 |

VII | 10 |

VIII | 11 |

IX | 14 |

X | 17 |

XII | 39 |

XLI | 102 |

XLII | 106 |

XLIII | 113 |

XLV | 118 |

XLVI | 121 |

XLVII | 126 |

XLVIII | 133 |

L | 136 |

XIV | 45 |

XV | 46 |

XVI | 48 |

XVII | 52 |

XVIII | 56 |

XIX | 57 |

XX | 59 |

XXII | 61 |

XXIII | 66 |

XXIV | 75 |

XXVI | 80 |

XXVIII | 82 |

XXIX | 83 |

XXX | 84 |

XXXI | 87 |

XXXII | 88 |

XXXIII | 92 |

XXXIV | 95 |

XXXV | 96 |

XXXVI | 98 |

XXXVIII | 99 |

XL | 100 |

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### Common terms and phrases

algorithm alternant code Aq(n binary BCH code binary code binary cyclic code binary Golay code binary Hamming code binary linear code called channel code of length code over F code with parameters codeword coding theory columns of H construct coordinates Corollary coset leader cyclic code cyclotomic cosets d]-linear code decoding rule defined Definition denote designed distance dimension divisor encoding equivalent error pattern Example Exercise exists finite field Fq[x Goppa code Hamming code Hence inner product irreducible polynomial Lemma Let q linearly independent matrix H MDS code minimal polynomial minimum distance narrow-sense binary BCH nearest neighbour decoding nonzero codeword obtain ord(a parity-check matrix Plotkin bound polynomial f(x positive integer prime primitive element Proof Proposition q-ary quadratic nonresidue quadratic residue modulo received word Reed-Muller codes Remark root rows of G satisfies self-dual Show subset subspace Suppose syndrome look-up table ternary Theorem vector space wt(x