## Error Correction Coding: Mathematical Methods and AlgorithmsAn unparalleled learning tool and guide to error correction coding Error correction coding techniques allow the detection and correction of errors occurring during the transmission of data in digital communication systems. These techniques are nearly universally employed in modern communication systems, and are thus an important component of the modern information economy. Error Correction Coding: Mathematical Methods and Algorithms provides a comprehensive introduction to both the theoretical and practical aspects of error correction coding, with a presentation suitable for a wide variety of audiences, including graduate students in electrical engineering, mathematics, or computer science. The pedagogy is arranged so that the mathematical concepts are presented incrementally, followed immediately by applications to coding. A large number of exercises expand and deepen students' understanding. A unique feature of the book is a set of programming laboratories, supplemented with over 250 programs and functions on an associated Web site, which provides hands-on experience and a better understanding of the material. These laboratories lead students through the implementation and evaluation of Hamming codes, CRC codes, BCH and R-S codes, convolutional codes, turbo codes, and LDPC codes. This text offers both "classical" coding theory-such as Hamming, BCH, Reed-Solomon, Reed-Muller, and convolutional codes-as well as modern codes and decoding methods, including turbo codes, LDPC codes, repeat-accumulate codes, space time codes, factor graphs, soft-decision decoding, Guruswami-Sudan decoding, EXIT charts, and iterative decoding. Theoretical complements on performance and bounds are presented. Coding is also put into its communications and information theoretic context and connections are drawn to public key cryptosystems. Ideal as a classroom resource and a professional reference, this thorough guide will benefit electrical and computer engineers, mathematicians, students, researchers, and scientists. |

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### Contents

A Context for Error Correction Coding | 2 |

Groups and Vector Spaces | 62 |

Linear Block Codes | 83 |

Cyclic Codes Rings and Polynomials | 113 |

A Linear Feedback Shift Registers | 154 |

Polynomial Division and Linear Feedback Shift Registers | 161 |

Rudiments of Number Theory and Algebra | 171 |

A How Many Irreducible Polynomials Are There? | 218 |

Bounds on Codes | 406 |

10Bursty Channels Interleavers and Concatenation | 425 |

SoftDecision Decoding Algorithms | 439 |

Convolutional Codes | 452 |

Trellis Coded Modulation | 535 |

Turbo Codes | 583 |

LowDensity ParityCheck Codes | 634 |

Decoding Algorithms on Graphs | 680 |

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### Common terms and phrases

AWGN BCH codes BCJR algorithm Berlekamp–Massey algorithm binary bit error block code bound BPSK channel check matrix code of length codeword coefficients column complexity computed convolutional codes corresponding coset cyclic code decoding algorithm defined Definition denote Determine dmin elements encoder equation error correction error locator polynomial error pattern Euclidean algorithm Example factor graph field function Gaussian GF(q Hamming code Hamming distance input bits integer interleaver interpolation iterations lattice LDPC codes Lemma LFSR linear linear code log likelihood minimal polynomials minimum distance modulo monomial multiplication nonzero obtained operation orthogonal output parity check matrix path metric points probability of error produces Proof Reed–Solomon codes representation represented roots sequence shift Show shown in Figure signal constellation symbols Table Tanner graph Theorem transmitted trellis turbo codes values variable node Viterbialgorithm weight zero