Advertisement

On the q-ary image of cyclic codes

  • Jørn M. Jensen
Conference paper
  • 136 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1255)

Abstract

In this paper we review some of the properties of constacyclic codes and their q-ary images. It is demonstrated that some of these both have a very simple encoder and can be decoded using standard equipment. Next we find all constacyclic codes whose q-ary images are by a Tschirnhaus transformation isomorphic to shortened cyclic codes. It turns out that this very large class also includes some cyclic codes. As an example the binary image of all cyclic MDS codes of length 2m+1 are isomorphic to shortened cyclic codes.

Keywords

Cyclic Code Minimal Polynomial Generator Polynomial Splitting Field Constacyclic 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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. Hanan and F. P. Palermo, “On cyclic codes for multi-phase data transmission system,” J. Soc. Indust. Appl. Math., vol. 12, no. 4, pp. 794–804, Dec. 1964.Google Scholar
  2. 2.
    F. J. MacWilliams, “On binary cyclic codes which are also cyclic codes over GF(2S),” SIAM J. Appl. Math., vol. 19, no. 1, pp. 75–95, July 1970.Google Scholar
  3. 3.
    P. Rabizzoni, “Démultipliés de codes sur une extension de \(\mathbb{F}_q\),” Thèse de Doctorat de 3-ième cycle, Université de Provence, France, 1983.Google Scholar
  4. 4.
    C. Mouaha, “Codes linéares sur un corps fini déduits de codes sur une extension,” Thèse de Doctorate de 3-ième cycle, Faculté des Sciences de Luminy, Luminy, France, 1988.Google Scholar
  5. 5.
    D. A. Leonard, “Linear cyclic codes of wordlength v over GF(q s) which are also linear cyclic codes of wordlength sv over GF(q),” Designs, Codes Cryptogr., vol. 1, no. 2, pp. 183–189, 1991.Google Scholar
  6. 6.
    G. Séguin, “A counter-example to a recent result on the q-ary image of a q s-ary cyclic code,” Designs, Codes Cryptogr., vol. 4, no. 2, pp. 171–175, 1994.Google Scholar
  7. 7.
    G. E. Séguin, “The q-ary Image of a q m-ary Cyclic Code,” IEEE Trans. Inform. Theory, vol. 41, pp. 387–399, March 1995.Google Scholar
  8. 8.
    E. R. Berlekamp, Algebraic Coding Theory. Laguna Hills, CA: Aegean Park Press, 1984.Google Scholar
  9. 9.
    A. Krishna and D. V. Sarwate, “Pseudocyclic maximum-distance-seperable codes,” IEEE Trans. Inform. Theory, vol. 36, pp. 880–884. July 1990.Google Scholar
  10. 10.
    J. M. Jensen, “A Class of Constacyclic Codes,” IEEE Trans. Inform. Theory, vol. 40, pp. 951–954, May 1994.Google Scholar
  11. 11.
    J. M. Jensen, “Cyclic concatenated codes with constacyclic outer codes,” IEEE Trans. Inform. Theory, vol. 38, pp. 950–959, May 1992.Google Scholar
  12. 12.
    J. P. Pedersea and C. Dahl, “Classification of pseudo-cyclic MDS codes,” IEEE Trans. Inform. Theory, vol. 37, pp. 365–370, March 1991.Google Scholar
  13. 13.
    J. M. Jensen, “On Decoding Doubly Extended Reed-Solomon Codes,” in Proc. IEEE Int. Symp. Inform. Theory, p. 280, (Canada,) Sept. 17–22, 1995.Google Scholar
  14. 14.
    L. Rédei, Algebra, vol. 1. New York: Pergamon Press, 1967.Google Scholar
  15. 15.
    J. M. Jensen, manuscript in preparation.Google Scholar
  16. 16.
    W. W. Peterson and E. J. Weldon, Jr., Error-Correcting Codes, 2nd ed., Cambridge, BA: MIT Press, 1972.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Jørn M. Jensen
    • 1
  1. 1.Department of MathematicsTechnical University of DenmarkLyngbyDenmark

Personalised recommendations