Construction of Large Constant Dimension Codes with a Prescribed Minimum Distance

  • Axel Kohnert
  • Sascha Kurz
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5393)


In this paper we construct constant dimension codes with prescribed minimum distance. There is an increased interest in subspace codes in general since a paper [13] by Kötter and Kschischang where they gave an application in network coding. There is also a connection to the theory of designs over finite fields. We will modify a method of Braun, Kerber and Laue [7] which they used for the construction of designs over finite fields to construct constant dimension codes. Using this approach we found many new constant dimension codes with a larger number of codewords than previously known codes. We finally give a table of the best constant dimension codes we found.


network coding q-analogue of Steiner systems subspace codes 


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  1. 1.
    Ahlswede, R., Aydinian, H.K., Khachatrian, L.H.: On perfect codes and related concepts. Des. Codes Cryptography 22(3), 221–237 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Betten, A., Braun, M., Fripertinger, H., Kerber, A., Kohnert, A., Wassermann, A.: Error-correcting linear codes. Classification by isometry and applications. With CD-ROM. In: Algorithms and Computation in Mathematics 18, p. xxix, 798. Springer, Berlin (2006)Google Scholar
  3. 3.
    Betten, A., Kerber, A., Kohnert, A., Laue, R., Wassermann, A.: The discovery of simple 7-designs with automorphism group PΓ L(2,32). In: Giusti, M., Cohen, G., Mora, T. (eds.) AAECC 1995. LNCS, vol. 948, pp. 131–145. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  4. 4.
    Braun, M.: Construction of linear codes with large minimum distance. IEEE Transactions on Information Theory 50(8), 1687–1691 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Braun, M., Kohnert, A., Wassermann, A.: Optimal linear codes from matrix groups. IEEE Transactions on Information Theory 51(12), 4247–4251 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Braun, M.: Some new designs over finite fields. Bayreuther Math. Schr. 74, 58–68 (2005)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Braun, M., Kerber, A., Laue, R.: Systematic construction of q-analogs of t-(v,k,λ)-designs. Des. Codes Cryptography 34(1), 55–70 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Braun, M., Kohnert, A., Wassermann, A.: Construction of (n,r)-arcs in PG(2,q). Innov. Incidence Geom. 1, 133–141 (2005)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Drudge, K.: On the orbits of Singer groups and their subgroups. Electronic Journal Comb. 9(1), 10 p. (2002)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Etzion, T., Silberstein, N.: Construction of error-correcting codes for random network coding (submitted, 2008) (in arXiv 0805.3528)Google Scholar
  11. 11.
    Etzion, T., Vardy, A.: Error-Correcting codes in projective space. In: ISIT Proceedings, 5 p. (2008)Google Scholar
  12. 12.
    Gadouleau, M., Yan, Z.: Constant-rank codes and their connection to constant-dimension codes (submitted, 2008) (in arXiv 0803.2262)Google Scholar
  13. 13.
    Kötter, R., Kschischang, F.: Coding for errors and erasures in random network coding. IEEE Transactions on Information Theory 54(8), 3579–3391 (2008)Google Scholar
  14. 14.
    Kramer, E.S., Mesner, D.M.: t-designs on hypergraphs. Discrete Math. 15, 263–296 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Maruta, T., Shinohara, M., Takenaka, M.: Constructing linear codes from some orbits of projectivities. Discrete Math. 308(5-6), 832–841 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Niskanen, S., Östergård, P.R.J.: Cliquer user’s guide, version 1.0. Technical Report T48, Communications Laboratory, Helsinki University of Technology, Espoo, Finland (2003)Google Scholar
  17. 17.
    Schwartz, M., Etzion, T.: Codes and anticodes in the Grassman graph. J. Comb. Theory, Ser. A 97(1), 27–42 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Silberstein, N.: Coding theory in projective space. Ph.D. proposal (2008) (in arXiv 0805.3528)Google Scholar
  19. 19.
    Thomas, S.: Designs over finite fields. Geom. Dedicata 24, 237–242 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Thomas, S.: Designs and partial geometries over finite fields. Geom. Dedicata 63(3), 247–253 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Tonchev, V.D.: Quantum codes from caps. Discrete Math. (to appear, 2008)Google Scholar
  22. 22.
    Wassermann, A.: Lattice point enumeration and applications. Bayreuther Math. Schr. 73, 1–114 (2006)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Xia, S.-T., Fu, F.-W.: Johnson type bounds on constant dimension codes (submitted, 2007) (in arXiv 0709.1074)Google Scholar
  24. 24.
    Zwanzger, J.: A heuristic algorithm for the construction of good linear codes. IEEE Transactions on Information Theory 54(5), 2388–2392 (2008)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Axel Kohnert
    • 1
  • Sascha Kurz
    • 1
  1. 1.Department of MathematicsUniversity of BayreuthBayreuthGermany

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