Skip to main content
SpringerLink
Log in
Book cover

International Symposium on Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes

AAECC 1999: Applied Algebra, Algebraic Algorithms and Error-Correcting Codes pp 160–169Cite as

Linear Codes and Rings of Matrices

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1719))

Abstract

We investigate the role of matrix rings in coding theory. Using these rings we introduce an embedding technique which enables us to give new interpretations for some previous results about algebraic representations of some prominent linear codes.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. F.W. Anderson and K. R. Fuller, Rings and categories of modules, Springer Verlag New York, Berlin, Heidelberg, 1992.

    MATH  Google Scholar 

  2. C. Bachoc, Applications of coding theory to the construction of modular lattices, J. Combin. Theory, Series A 78 (1997), 92–119.

    MATH  MathSciNet  Google Scholar 

  3. A. Bonnecaze and P. Udaya, Cyclic codes and self-dual codes over F 2 + ūF 2, Preprint, 1997.

    Google Scholar 

  4. N. Bourbaki, Elements of mathematics, ch. Commutative algebra, Addison-Wesley, Reading, MA, 1972.

    Google Scholar 

  5. A. E. Brouwer, Bounds on the size of linear codes, Handbook of coding theory, Vol. I, II, North-Holland, Amsterdam, 1998, pp. 295–461.

    MathSciNet  Google Scholar 

  6. P. J. Cameron and J. H. van Lint, Designs, graphs, codes and their links, Cambridge University Press, Cambridge, 1991.

    MATH  Google Scholar 

  7. D. Y. Goldberg, Reconstructing the ternary Golay code, J. Combin. Theory Ser. A 42 (1986), no. 2, 296–299.

    MATH  Google Scholar 

  8. M. Greferath, Cyclic codes over finite rings, Discrete Math. 177 (1997), no. 1-3, 273–277.

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  10. N. Jacobson, Basic Algebra II, W. H. Freeman and Company New York, 1989.

    MATH  Google Scholar 

  11. J. Lambek, Lectures on rings and modules, Blaisdell Publ. Co., Waltham, Massachusetts-Toronto-London, 1966.

    MATH  Google Scholar 

  12. A. A. Nechaev, Similarity of matrices over commutative Artinian rings, J. of Soviet. Math. 33 (1986), 1221–1237.

    MATH  Google Scholar 

  13. A. A. Nechaev, Kerdock codes in a cyclic form, Discrete Math. Appl. 1 (1991), 365–384.

    MATH  MathSciNet  Google Scholar 

  14. G. Pasquier, The binary GOLAY code obtained from an extended cyclic code over F 8, European J. Combin. 1 (1980), no. 4, 369–370.

    MathSciNet  Google Scholar 

  15. V. S. Pless, W. C. Huffman, and R. A. Brualdi( eds.), Handbook of coding theory. Vol. I, II, North-Holland, Amsterdam, 1998.

    Google Scholar 

  16. J. Wolfmann, A new construction of the binary Golay code (24, 12, 8) using a group algebra over a finite field, Discrete Math. 31 (1980), no. 3, 337–338.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. AT&T Labs-Research, 180 Park Ave, Florham Park, NJ, 07932, USA

    M. Greferath & S. E. Schmidt

Authors
  1. M. Greferath
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. E. Schmidt
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Electrical Engineering, University of Hawaii, 2540 Dole St., # 483, Honolulu, HI, 96822, USA

    Marc Fossorier  & Shu Lin  & 

  2. Institute of Industrial Science, University of Tokyo, 7-22-1, Roppongi, Minato-ku, Tokyo, 106, Japan

    Hideki Imai

  3. AAECC/IRIT, Université Paul Sabatier, 118, Route de Narbonne, F31067, Toulouse, Cedex, France

    Alain Poli

Rights and permissions

Reprints and permissions

Copyright information

© 1999 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Greferath, M., Schmidt, S.E. (1999). Linear Codes and Rings of Matrices. In: Fossorier, M., Imai, H., Lin, S., Poli, A. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 1999. Lecture Notes in Computer Science, vol 1719. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-46796-3_16

Download citation

  • DOI: https://doi.org/10.1007/3-540-46796-3_16

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-66723-0

  • Online ISBN: 978-3-540-46796-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation