Designs, Codes and Cryptography

, Volume 37, Issue 2, pp 211–214 | Cite as

A Family of Binary (t, m,s)-Nets of Strength 5

  • Jürgen Bierbrauer
  • Yves Edel


(t,m,s)-Nets were defined by Niederreiter [Monatshefte fur Mathematik, Vol. 104 (1987) pp. 273–337], based on earlier work by Sobol’ [Zh. Vychisl Mat. i mat. Fiz, Vol. 7 (1967) pp. 784–802], in the context of quasi-Monte Carlo methods of numerical integration. Formulated in combinatorial/coding theoretic terms a binary linear (mk,m,s)2-net is a family of ks vectors in F 2 m satisfying certain linear independence conditions (s is the length, m the dimension and k the strength: certain subsets of k vectors must be linearly independent). Helleseth et al. [5] recently constructed (2r−3,2r+2,2 r −1)2-nets for every r. In this paper, we give a direct and elementary construction for (2r−3,2r+2,2 r +1)2-nets based on a family of binary linear codes of minimum distance 6.


Data Structure Information Theory Minimum Distance Linear Code Discrete Mathematic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bierbrauer, J. 2001The theory of cyclic codes and a generalization to additive codesDesigns, Codes and Cryptography25189206Google Scholar
  2. 2.
    Bierbrauer, J., Edel, Y. 1997Construction of digital nets from BCH-codes. Monte Carlo and Quasi-Monte Carlo Methods 1996Lecture Notes in Statistics127221231Google Scholar
  3. 3.
    Bierbrauer, J., Edel, Y., Ch. Schmid, W. 2002Coding-theoretic constructions for tms-nets and ordered orthogonal arraysJournal of Combinatorial Designs10403418CrossRefGoogle Scholar
  4. 4.
    Edel, Y., Bierbrauer, J. 2001Families of ternary (t,m,s)-nets related to BCH-codesMonatshefte für Mathematik13299103CrossRefGoogle Scholar
  5. 5.
    Helleseth, T., Kløve, T., Levenshtein, V. 2003Hypercubic 4-and 5-designs from double-error-correcting BCH codesDesigns, Codes and Cryptography28265282Google Scholar
  6. 6.
    Niederreiter, H. 1987Point sets and sequences with small discrepancyMonatshefte für Mathematik104273337CrossRefGoogle Scholar
  7. 7.
    I. M. Sobol’, Distribution of points in a cube and the approximate evaluation of integrals (in Russian), Zh. Vychisl.Mat. i Mat. Fiz, Vol. 7 (1967) pp. 784–802. English Translation in USSR Computational Mathematics and Mathematical Physics, Vol. 7 (1967) pp. 86–112.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of Mathematical SciencesMichigan Technological UniversityHoughtonUSA
  2. 2.Mathematisches Institut der UniversitätHeidelbergGermany

Personalised recommendations