An Extension of the Brouwer-Zimmermann Minimum Weight Algorithm

Conference paper
Part of the CIM Series in Mathematical Sciences book series (CIMSMS, volume 3)


We study the algorithm for computing the minimum weight of a linear code that was invented by A. Brouwer and later extended by K.-H. Zimmermann. We show that matroid partitioning algorithms can be used to efficiently find a favourable (and sometimes best possible) sequence of information sets on which the Brouwer-Zimmermann minimum weight algorithm operates.


Linear code Minimum weight Brouwer-Zimmermann algorithm 


  1. 1.
    Betten, A., Fripertinger, H., Kerber, A., Wassermann, A., Zimmermann, K.-H.: Codierungstheorie–Konstruktion und Anwendung Linearer Codes. Springer, Berlin (1998)MATHGoogle Scholar
  2. 2.
    Betten, A., Braun, M., Fripertinger, H., Kerber, A., Kohnert, A., Wassermann, A.: Error-Correcting Linear Codes. Springer, Berlin/New York (2006)MATHGoogle Scholar
  3. 3.
    Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24, 235–265 (1997)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cunningham, W.H.: Improved bounds for matroid partition and intersection algorithms. SIAM J. Comput. 15, 948–957 (1986)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Edmonds, J.: Minimum partition of a matroid into independent subsets. J. Res. Nat. Bur. Standards Sect. B 69B, 67–72 (1965)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Grassl, M.: Searching for linear codes with large minimum distance. In: Bosma, W., Cannon, J. (eds.) Discovering Mathematics with Magma – Reducing the Abstract to the Concrete, pp. 287–313. Springer, Berlin/New York (2006)CrossRefGoogle Scholar
  7. 7.
    Kiermaier, M., Wassermann, A.: Minimum weights and weight enumerators of \(\mathbb{Z}_{4}\)-linear quadratic residue codes. IEEE Trans. Inf. Theory 58, 4870–4883 (2012)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Lawler, E.L.: Combinatorial Optimization: Networks and Matroids. Holt, Rinehart and Winston, New York (1976)MATHGoogle Scholar
  9. 9.
    MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam/New York (1977)MATHGoogle Scholar
  10. 10.
    Oxley, J.: Matroid Theory, 2nd edn. Oxford University Press, Oxford/New York (2011)CrossRefMATHGoogle Scholar
  11. 11.
    Vardy, A.: The intractability of computing the minimum distance of a code. IEEE Trans. Inf. Theory 43, 1757–1766 (1997)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of MathematicsSimon Fraser UniversityBurnabyCanada

Personalised recommendations