Skip to main content
SpringerLink
Log in

Certain self-dual codes over ℤ4 and the odd Leech lattice

  • Conference paper
  • First Online:
Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC 1997)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1255))

Abstract

Recently, alternative constructions of the Leech lattice and the shorter Leech lattice have been discovered using self-dual codes over ℤ4. In this paper, we provide a classification of length 24 double circulant Type I codes over ℤ4 with minimum Euclidean weight 12. These codes determine (via Construction A4) the odd Leech lattice, which is a unique 24-dimensional odd unimodular lattice with minimum norm 3.

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. E. Bannai, S.T. Dougherty, M. Harada and M. Oura, Type II codes, even unimodular lattices and invariant rings, (in preparation).

    Google Scholar 

  2. A. Bonnecaze, P. Solé and A.R. Calderbank, Quaternary quadratic residue codes and unimodular lattices, IEEE Trans. Inform. Theory Vol. 41 (1995) pp. 366–377.

    Google Scholar 

  3. A. Bonnecaze, P. Solé, C. Bachoc and B. Mourrain, Type II codes over ℤ4, IEEE Trans. Inform. Theory, (to appear).

    Google Scholar 

  4. A.R. Calderbank and N.J.A. Sloane, Double circulant codes over ℤ4 and even unimodular lattices, J. Alg. Combin., (to appear).

    Google Scholar 

  5. J.H. Conway and N.J.A. Sloane, Self-dual codes over the integers modulo 4, J. Combin. Theory Ser. A Vol. 62 (1993) pp. 30–45.

    Google Scholar 

  6. J.H. Conway and N.J.A. Sloane, Sphere Packing, Lattices and Groups (2nd ed.), Springer-Verlag, New York, 1993.

    Google Scholar 

  7. S.T. Dougherty, T.A. Gulliver and M. Harada, Type II self-dual codes over finite rings and even unimodular lattices, (submitted to J. Alg. Combin.).

    Google Scholar 

  8. T.A. Gulliver and M. Harada, Extremal double circulant Type II codes over ℤ4 and construction of 5-(24,10,36) designs, (submitted to Discrete Math.).

    Google Scholar 

  9. M. Harada, New extremal Type II codes over ℤ4, (submitted to Des. Codes and Cryptogr.).

    Google Scholar 

  10. V. Pless, P. Solé and Z. Qian, Cyclic self-dual ℤ4-codes, Finite Fields and Their Appl. Vol. 3 (1997) pp. 48–69.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electrical and Electronic Engineering, University of Canterbury, Private Bag 4800, Christchurch, New Zealand

    T. Aaron Gulliver

  2. Department of Mathematical Sciences, Yamagata University, 990, Yamagata, Japan

    Masaaki Harada

Authors
  1. T. Aaron Gulliver
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Masaaki Harada
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Teo Mora Harold Mattson

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Gulliver, T.A., Harada, M. (1997). Certain self-dual codes over ℤ4 and the odd Leech lattice. In: Mora, T., Mattson, H. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 1997. Lecture Notes in Computer Science, vol 1255. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63163-1_10

Download citation

  • DOI: https://doi.org/10.1007/3-540-63163-1_10

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-63163-7

  • Online ISBN: 978-3-540-69193-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation