Skip to main content
SpringerLink
Log in

Information sets in abelian codes: defining sets and Groebner basis

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

In Bernal and Simón (IEEE Trans Inf Theory 57(12):7990–7999, 2011) we introduced a technique to construct information sets for every semisimple abelian code by means of its defining set. This construction is a non trivial generalization of that given by Imai (Inf Control 34:1–21, 1977) in the case of binary two-dimensional cyclic (TDC) codes. On the other hand, Sakata (IEEE Trans Inf Theory IT-27(5):556–565, 1981) showed a method for constructing information sets for binary TDC codes based on the computation of Groebner basis which agrees with the information set obtained by Imai. Later, Chabanne (IEEE Trans Inf Theory 38(6):1826–1829, 1992) presents a generalization of the permutation decoding algorithm for binary abelian codes by using Groebner basis, and as a part of his method he constructs an information set following the same ideas introduced by Sakata. In this paper we show that, in the general case of q-ary multidimensional abelian codes, both methods, that based on Groebner basis and that defined in terms of the defining sets, also yield the same information set.

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

Access this article

Log in via an institution

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. Berman S.D.: Semisimple cyclic and abelian codes. Cybernetics 3(3), 21–30 (1967)

    MathSciNet  Google Scholar 

  2. Bernal J.J., Simón J.J.: Information sets for abelian codes. In: 2010 IEEE Information Theory Workshop—ITW 2010 Dublin (2010).

  3. Bernal J.J., Simón J.J.: Information sets from defining sets in abelian codes. IEEE Trans. Inf. Theory 57(12), 7990–7999 (2011)

    Article  Google Scholar 

  4. Buchberger B.: Groebner bases: an algorithm method in polynomial ideal theory. In: Bose, (ed) Recent Trends in Multidimensional System Theory, Reidel, Dordrecht (1985)

    Google Scholar 

  5. Camion P.: Abelian codes. MRC Tech. Sum. Rep. no. 1059, University of Wisconsin, Madison (1970).

  6. Chabanne H.: Permutation decoding of abelian codes. IEEE Trans. Inf. Theory 38(6), 1826–1829 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  7. Coffey J.T., Goodman R.M.: The complexity of information set decoding. IEEE Trans. Inf. Theory 36(5), 1031–1037 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cox D., Little J., O’Shea D.: Ideals, Varieties, and Algorithms. Springer, New York (1992)

    Book  MATH  Google Scholar 

  9. Huffman W.C.: Codes and groups. In: Pless, V.S., Huffman, W.C., Brualdi, R.A. (eds) Handbook of Coding Theory, vol. II., pp. 1345–1440. North-Holland, Amsterdam (1998)

    Google Scholar 

  10. Imai H.: A theory of two-dimensional cyclic codes. Inf. Control 34, 1–21 (1977)

    Article  MATH  Google Scholar 

  11. Imai H.: Multivariate polynomials in coding theory. In: AAECC-2, LNCS, vol. 228, pp. 36–60 (1986).

  12. Key J.D., McDonough T.P., Mavron V.C.: Partial permutation decoding for codes from finite planes. Eur. J. Comb. 26, 665–682 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  13. Key J.D., McDonough T.P., Mavron V.C.: Information sets and partial permutation decoding for codes from finite geometries. Finite Fields Appl. 12, 232–247 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  14. Kroll H.J., Vincenti R.: PD-sets for the codes related to some classical varieties. Discret. Math. 301, 89–105 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  15. MacWilliams F.J.: Permutation decoding of systematic codes. Bell. Syst. Tech. J. 43, 485–505 (1964)

    Article  MATH  Google Scholar 

  16. MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1983)

    Google Scholar 

  17. Peters C.: Information-set decoding for linear codes over F q . In: 2010 Third International Workshop, PQCrypto, Darmstadt. Springer, Berlin.

  18. Pless, V.S., Huffman, W.C., Brualdi, R.A. (eds): Handbook of Coding Theory, vol. I. North-Holland, Amsterdam (1998)

    Google Scholar 

  19. Poli A.L., Huguet L.: Codes Correcteurs. Masson, Paris (1988)

    Google Scholar 

  20. Prange E.: The use of information sets in decoding cyclic codes. IRE Trans. IT-8, S5–S9 (1962)

    MathSciNet  Google Scholar 

  21. Sakata S.: On determining the independent point set for doubly periodic arrays and encoding two-dimensional cyclic codes and their duals. IEEE Trans. Inf. Theory IT-27(5), 556–565 (1981)

    Article  MathSciNet  Google Scholar 

  22. Seneviratne P.: Partial permutation decoding for the first-order Reed–Muller codes. Discret. Math. 309, 1967–1970 (2009)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemáticas, Universidad de Murcia, Murcia, Spain

    José Joaquín Bernal & Juan Jacobo Simón

Authors
  1. José Joaquín Bernal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Juan Jacobo Simón
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Juan Jacobo Simón.

Additional information

This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding Theory and Applications”.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bernal, J.J., Simón, J.J. Information sets in abelian codes: defining sets and Groebner basis. Des. Codes Cryptogr. 70, 175–188 (2014). https://doi.org/10.1007/s10623-012-9735-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-012-9735-x

Keywords

Mathematics Subject Classification (2010)

Search

Navigation