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A First Look at Block Coders

  • Emilio Sanvicente
Chapter

Abstract

This chapter shows how the code and the coder can be defined by two matrices: the parity-check matrix and the generator matrix. It is explained how the correction/detection capability of the code depends on the linear independence of sets of columns of the parity-check matrix. The arithmetic is performed in finite sets, called Galois fields. Three introductory examples of Galois fields of two, three, and four elements are presented. The field with 11 elements is also easy to obtain, and it is used to decode erasures. Erasure decoding is then applied to the recovery of lost packets in data networks using Vandermonde and Cauchy matrices defined in the field with eight elements constructed in Appendix C. The chapter finishes deriving the Hamming bound, presenting the family of Hamming codes, and explaining how to obtain new codes from a given code. Appendix D is a review of the concept of rank of a matrix.

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Emilio Sanvicente
    • 1
  1. 1.Former Professor of Electrical Engineering, School of Telecommunication EngineeringPolytechnic University of CataloniaBarcelonaSpain

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