Skip to main content
SpringerLink
Log in

Independence of vectors in codes over rings

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

Abstract

We study codes over Frobenius rings. We describe Frobenius rings via an isomorphism to the product of local Frobenius rings and use this decomposition to describe an analog of linear independence. Special attention is given to codes over principal ideal rings and a basis for codes over principal ideal rings is defined. We prove that a basis exists for any code over a principal ideal ring and that any two basis have the same number of vectors.

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. Dougherty S.T., Kim J.-L., Kulosman H.: MDS codes over finite principal ideal rings. Des. Codes Cryptogr. 48(3). doi:10.1007/s10623-008-9215-5.

  2. Faith C.: Algebra II: Ring Theory. Grundlehren Math. Wiss, Bd.191. Springer, Berlin (1976)

    Google Scholar 

  3. Hammons A.R. Jr., Kumar P.V., Calderbank A.R., Sloane N.J.A., Solé P.: The Z 4 linearity of Kerdock, Preparata, Goethals and related codes. IEEE Trans. Inform. Theory 40, 301–319 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  4. Horimoto H., Shiromoto K.: A Singleton bound for linear codes over quasi-Frobenius rings. In: Proceedings of the 13th International Symposium on Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes, Hawaii (USA), pp. 51–52 (1999).

  5. MacWilliams F.J., Sloane N.J.A.: The theory of error-correcting codes. North-Holland, Amsterdam (1977)

    MATH  Google Scholar 

  6. McDonald D.: Finite rings with identity. Marcel Dekker, New York (1974)

    MATH  Google Scholar 

  7. Norton G.H., Sǎlǎgean A.: On the structure of linear and cyclic codes over a finite chain ring. Appl. Algebra Engrg. Comm. Comput. 10, 489–506 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  8. Park Y.H.: Modular independence and generator matrices for codes over Z m . Des. Codes Cryptogr. doi:10.1007/s10623-008-9220-8.

  9. Wood J.: Duality for modules over finite rings and applications to coding theory. Amer. J. Math. 121, 555–575 (1999)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Scranton, Scranton, PA, 18510, USA

    Steven T. Dougherty

  2. Department of Mathematics, Huazhong Normal University, Wuhan, Hubei, 430079, China

    Hongwei Liu

Authors
  1. Steven T. Dougherty
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hongwei Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Steven T. Dougherty.

Additional information

Communicated by J.D. Key.

Hongwei Liu is supported by the National Natural Science Foundation of China (10571067).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dougherty, S.T., Liu, H. Independence of vectors in codes over rings. Des. Codes Cryptogr. 51, 55–68 (2009). https://doi.org/10.1007/s10623-008-9243-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-008-9243-1

Keywords

Mathematics Subject Classifications (2000)

Search

Navigation