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Parity Check Matrix Approach to Linear Block Codes

  • Giovanni CancellieriEmail author
Chapter
Part of the Signals and Communication Technology book series (SCT)

Abstract

The parity check matrix can be assumed for an alternative description of a linear code. The relationship with the generator matrix is not univocal, except when such matrices are systematic. Upper triangular generator matrices and lower triangular parity check matrices are presented, showing the advantage of a clear recognition of information and control symbol positions. Error correction is outlined by means of properly processing single-error syndromes. A strict-sense time-invariant code in its parity check matrix is characterized by a unique parity check polynomial. The concept of dual code is discussed. Code puncturation and code shortening are interpreted as dual operations. Constant-length puncturation and constant-length shortening are described. Periodic parity check polynomials in lengthened cyclic codes are constructed. The parity check matrix of a modified lengthened cyclic code is derived. The difference between periodic and non-periodic parity check rows is stressed. H-extended cyclic codes and modified H-extended cyclic codes are introduced. Generalized repetition codes represent dual codes of generalized parity check codes. The whole family of Hamming codes is constructed by repeated lengthening operations. The whole family of simplex codes is constructed by repeated H-extensions. Direct product codes are described by means of their parity check matrix. An encoder circuit based on the parity check polynomial is depicted. Its state diagram is studied, and the trellis obtained from it is compared with the traditional generator trellis. Finally some considerations about the parity check matrix of nonbinary block codes are developed, with particular attention to their error correction capability.

Keywords

Block Code LDPC Code Parity Check Code Word Cyclic Code 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. Baldi M, Cancellieri G, Carassai A et al (2009) LDPC codes based on serially concatenated multiple parity—check codes. IEEE Comm Lett 13:142–144CrossRefGoogle Scholar
  2. Berlekamp ER (1965) On decoding Bose-Chauduri-Hocquenghem codes. IEEE Trans Inf Theory 11:577–580CrossRefzbMATHMathSciNetGoogle Scholar
  3. Clark GC, Cain JB (1983) Error correcting coding for digital communications. Plenum, New YorkGoogle Scholar
  4. De Bruijn NG (1946) A combinatorial problem. J London Math Soc 21:167–169MathSciNetGoogle Scholar
  5. Gorenstein DC, Zierler N (1961) A class of error correcting codes in p m symbols. J SIAM 9:207–214zbMATHMathSciNetGoogle Scholar
  6. Johannesson R, Zigangirov KS (1999) Fundamentals of convolutional coding. IEEE Press, New YorkCrossRefGoogle Scholar
  7. Klove T (2007) Codes for error detection. World Scientific, SingaporeGoogle Scholar
  8. MacWilliams FJ, Sloane NJA (1977) The theory of error correcting codes. North Holland, New YorkzbMATHGoogle Scholar
  9. Massey JL (1969) Shift-register synthesis and BCH decoding. IEEE Trans Inf Theory 15:122–127CrossRefzbMATHMathSciNetGoogle Scholar
  10. Peterson WW (1960) Encoding and error-correction procedures. IRE Trans Inf Theory 6:459–470CrossRefGoogle Scholar
  11. Rankin DM, Gulliver TA (2001) Single parity check product codes. IEEE Trans Commun 49:1354–1362CrossRefzbMATHGoogle Scholar
  12. Ryan WE, Lin S (2009) Channel codes: classical and modern. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  13. Sweeney P (2002) Error control coding. Wiley, New YorkGoogle Scholar
  14. Wicker SB (1995) Error control systems for digital communications and storage. Prentice-Hall, Englewood CliffsGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Information EngineeringPolytechnic University of MarcheAnconaItaly

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