Skip to main content
Log in

Structural properties and enumeration of quasi cyclic codes

  • Published:
Applicable Algebra in Engineering, Communication and Computing Aims and scope

Abstract

Given any finite fieldF q , an (N, K) quasi cyclic code is defined as aK dimensional linear subspace ofF N q which is invariant underT nfor some integern, 0 <nN, and whereT is the cyclic shift operator. Quasi cyclic codes are shown to be isomorphic to theF q [λ]-submodules ofF N q where the productμ(gl)·ν is naturally defined asμ 0 ν+μ 1νTn +...+μ m νT mnifμ(λ)= μ 0 1 +...+μ m λ m .In the case where (N/n, q)=1, all quasi cyclic codes are shown to be decomposable into the direct sum of a fixed number of indecomposable components called irreducible cyclicF q [λ]-submodules providing for the complete characterisation and enumeration of some subclasses of quasi cyclic codes including the cyclic codes, the quasi cyclic codes with a cyclic basis, the maximal and the irreducible ones. Finally a general procedure is presented which allows for the determination and characterisation of the dual of any quasi cyclic code.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bourbaki, N.: Modules sur les anneaux principaux. In: Algèbre, Chap. VII, Paris: Hermann 1959

    Google Scholar 

  2. Camion, P.: Codes correcteurs d'erreurs. Revue du CETHEDEC, Numéro spécial, 3ième trimestre 1966

  3. Curtis, C. W., Reiner, I.: Representation Theory of Finite Groups and Associative Algebras. New York: John Wiley 1962

    Google Scholar 

  4. Drolet, G.: Sur les codes quasi-cycliques à base cyclique. Thèse de Ph.D., Université Laval, Québec, Canada, June 1990

    Google Scholar 

  5. Jacobson, N.: Lectures in Abstract Algebra, Vol II. New York: D. Van Nostrand 1953

    Google Scholar 

  6. Kasami, T.: A Gilbert-Varshamov Bound for Quasi Cyclic Codes of Rate 1/2. IEEE Trans. Inf. Theory, p. 679, 1974

  7. MacWilliams, F. J. Sloane, N. J. A.: The Theory of Error-Correcting Codes. New York: North Holland 1978

    Google Scholar 

  8. Massey, J. L.: Error Bounds for Tree Codes, Trellis Codes and Convolutional Codes with Encoding and Decoding Procedures. In: Coding and Complexity, Longo G. (ed) Berlin, Heidelberg New York: Springer 1976

    Google Scholar 

  9. Séguin, G. E., Huynh, H. T.: Quasi-Cyclic Codes: A Study. Report published by the Laboratoire de Radiocommunications et de Traitement du Signal, Université Laval, Québec, Canada, 1985

    Google Scholar 

  10. Solomon, G., van Tilborg, H. C. A.: A Connection between Block and Convolutional Codes. SIAM J. Appl. Math.,37, No(2), 358–369 (1989)

    Google Scholar 

  11. Viterbi, A. J.: Convolutional Codes and their Performance in Communication Systems. IEEE Trans. Comm. Tech.COM-19, 751–772 (1971)

    Google Scholar 

  12. Zigangirov, K. SR.: Some Sequential Decoding Procedures. Prob. Peredachi Inform.2, 13–25 (1966)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Conan, J., Séguin, G. Structural properties and enumeration of quasi cyclic codes. AAECC 4, 25–39 (1993). https://doi.org/10.1007/BF01270398

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01270398

Keywords

Navigation