Algebraic Codes in the Frequency Domain

  • Richard E. Blahut
Part of the International Centre for Mechanical Sciences book series (CISM)


Communication theory and signal processing are closely related subjects that have been developed largely by engineers. Analysis and synthesis problems in these fields depend heavily on reasoning in the frequency domain. Thus, in the study of real- or complex-valued signals, the Fourier transform plays a basic role. When the time variable is discrete, the discrete Fourier transform plays a parallel role. Accordingly, Fourier transforms and discrete Fourier transforms are among the major tools of engineers.


Cyclic Code Parity Frequency Information Symbol Parity Check Matrix Error Control 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.


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  1. 1.
    Pollard, J.M., “The Fast Fourier Transform in a Finite Field,” Mathematics of Computation, pp 365–374, Vol. 25, No. 114, April 1971.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Gore, W.C., “Transmitting Binary Symbols With Reed-Solomon Codes,” Proceedings of Princeton Conference on Information Sciences and Systems, Princeton, N.J., pp 495–497, 1973.Google Scholar
  3. 3.
    Micheison, A., “A Fast Transform in Some Galois Fields and an Application to Decoding Reed-Solomon Codes,” IEEE Abstracts of Papers — IEEE International Symposium on Information Theory, Ronneby, Sweden, 1976.Google Scholar
  4. 4.
    Lempel, A. and Winograd, S., “A New Approach to Error Correcting Codes,” IEEE Trans. Information Theory, Vol IT 23, pp 503–508, July 1977.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chien, R.T., and Choy, D.M., “Algebraic Generalization of BCH-Goppa-Helgert Codes,” IEEE Trans. Information Theory, Vol IT 21, pp 70–79, January 1975.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Mattson, H.F., and G. Solomon, “A New Treatment of Bose Chandhuri Codes,” J. Soc Indus. Appl. Math., 9, 4, 654–699, 1961.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Peterson, W.W., amd Weldon, E.J. Jr., Error Correcting Codes 2nd Ed., MIT Press, 1972.zbMATHGoogle Scholar
  8. 8.
    MacWilliams, F.J., and Sloane, N.J.A., The Theory of Error Correcting Codes, North Holland, Amsterdam, 1977.zbMATHGoogle Scholar
  9. 9.
    Berlekamp, E.R., Algebraic Coding Theory, Mc Graw Hill, New York, 1968.zbMATHGoogle Scholar
  10. 10.
    Chien, R.T., “A New Proof of the BCH Bound,” IEEE Trans. Information Theory, Vol IT 18, p. 541, July, 1972.Google Scholar
  11. 11.
    Massey, J.L., “Shift-Register Synthesis and BCH Decoding,” IEEE Trans. Information Theory, Vol IT 15, pp 122–127, 1969.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Berlekamp, E., “Long Primitive BCH Codes Have Distance d~2n lnR~ /log n,” IEEE Trans. Information Theory, Vol IT 18, pp 415–426, May 1972.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Helgert, H.H., “Alternant Codes,” Information and Control, pp 369–381, 1974.Google Scholar
  14. 14.
    Goppa, V.C., “A New Class of Linear Error-Correcting Codes” Probl. Peredach. Inform., Vol. 6, pp 24–30, September, 1970.MathSciNetzbMATHGoogle Scholar
  15. 15.
    Delsarte, P., “On Subfield Subcodes of Modified Reed-Solomon Codes,” IEEE Trans Information Theory, Vol IT 21, pp 575–576, September, 1975.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Elias, P., “Error Free Coding” IRE Trans. Information Theory, Vol IT 4, pp 29–37, 1954.MathSciNetGoogle Scholar
  17. 17.
    Burton, H.O., and eldon, E.J. Jr., “Cyclic Product Codes” IEEE Trans. Information Theory, Vol IT 11, pp 433–439, 1965.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lin, S. and Weldon, E.J. Jr., “Furthur Results on Cyclic Product Codes” IEEE Trans. Information Theory, Vol IT 16, pp 452–459, 1970.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Wolf, J.K., “Adding Two Information Symbols to Certain Nonbinary BCH Codes and Some Applications,” Bell Syst. Tech. J., pp 2405–2424, 1969.Google Scholar
  20. 20.
    Patterson, N.J., “The Algebraic Decoding of Goppa Codes,” IEEE Trans. Information Theory, Vol IT 21, pp 203–207, 1975.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Wien 1979

Authors and Affiliations

  • Richard E. Blahut
    • 1
  1. 1.Federal Systems DivisionInternational Business Machines CorporationOwegoUSA

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