An Extension of the Brouwer-Zimmermann Minimum Weight Algorithm

  • Petr Lisoněk
  • Layla Trummer
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 



Research of both authors was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). The authors thank Luis Goddyn for helpful comments and discussions.


  1. 1.
    Betten, A., Fripertinger, H., Kerber, A., Wassermann, A., Zimmermann, K.-H.: Codierungstheorie–Konstruktion und Anwendung Linearer Codes. Springer, Berlin (1998)zbMATHGoogle Scholar
  2. 2.
    Betten, A., Braun, M., Fripertinger, H., Kerber, A., Kohnert, A., Wassermann, A.: Error-Correcting Linear Codes. Springer, Berlin/New York (2006)zbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)zbMATHGoogle Scholar
  9. 9.
    MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam/New York (1977)zbMATHGoogle Scholar
  10. 10.
    Oxley, J.: Matroid Theory, 2nd edn. Oxford University Press, Oxford/New York (2011)CrossRefzbMATHGoogle 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

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