Designs, Codes and Cryptography

, Volume 87, Issue 2–3, pp 375–391 | Cite as

Classifying optimal binary subspace codes of length 8, constant dimension 4 and minimum distance 6

  • Daniel HeinleinEmail author
  • Thomas Honold
  • Michael Kiermaier
  • Sascha Kurz
  • Alfred Wassermann
Part of the following topical collections:
  1. Special Issue: Coding and Cryptography


We determine the maximum size \(A_2(8,6;4)\) of a binary subspace code of packet length \(v=8\), minimum subspace distance \(d=6\), and constant dimension \(k=4\) to be 257. There are two isomorphism types of optimal codes. Both of them are extended LMRD codes. In finite geometry terms, the maximum number of solids in \({\text {PG}}(7,2)\) mutually intersecting in at most a point is 257. The result was obtained by combining the classification of substructures with integer linear programming techniques. This result implies that the maximum size \(A_2(8,6)\) of a binary mixed-dimension subspace code of packet length 8 and minimum subspace distance 6 is 257 as well.


Network coding Constant-dimension codes Subspace distance Classification Integer linear programming 

Mathematics Subject Classification

51E20 94B65 05B25 51E23 



The authors would like to thank the High Performance Computing group of the University of Bayreuth for providing the excellent computing cluster and especially Bernhard Winkler for his support. This work was supported by the grants KU 2430/3-1, WA 1666/9-1—“Integer Linear Programming Models for Subspace Codes and Finite Geometry”—from the German Research Foundation and by Grant No. 61571006—“Research on Subspace Codes and Related Combinatorial Structures”—from the National Natural Science Foundation of China. The authors would like to thank the editor and the anonymous referees for their remarks improving the presentation of our results.


  1. 1.
    Beutelspacher A.: Partial spreads in finite projective spaces and partial designs. Math. Z. 145(3), 211–229 (1975).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Braun M., Etzion T., Ostergård P.R.J., Vardy A., Wassermann A.: Existence of \(q\)-analogs of Steiner systems. Forum Math. Pi 4, e7–14 (2016). Scholar
  3. 3.
    Delsarte P.: Bilinear forms over a finite field, with applications to coding theory. J. Comb. Theory Ser. A 25(3), 226–241 (1978).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Etzion T., Silberstein N.: Error-correcting codes in projective spaces via rank-metric codes and Ferrers diagrams. IEEE Trans. Inform. Theory 55(7), 2909–2919 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Etzion T., Silberstein N.: Codes and designs related to lifted MRD codes. IEEE Trans. Inform. Theory 59(2), 1004–1017 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Etzion T., Storme L.: Galois geometries and coding theory. Des. Codes Cryptogr. 78(1), 311–350 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Etzion T., Vardy A.: Error-correcting codes in projective space. IEEE Trans. Inform. Theory 57(2), 1165–1173 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gabidulin E.M.: Theory of codes with maximum rank distance. Probl. Pereda. Inform. 21(1), 3–16 (1985).MathSciNetzbMATHGoogle Scholar
  9. 9.
    Heinlein D., Kurz S.: Asymptotic bounds for the sizes of constant dimension codes and an improved lower bound. In: 5th International Castle Meeting on Coding Theory and Applications, pp. 1–30 (2017). arxiv:1703.08712.
  10. 10.
    Heinlein D., Kurz S.: An upper bound for binary subspace codes of length \(8\), constant dimension \(4\) and minimum distance \(6\). In: The Tenth International Workshop on Coding and Cryptography. (2017). arxiv:1705.03835.
  11. 11.
    Heinlein D., Kiermaier M., Kurz S., Wassermann A.: Tables of subspace codes. (2016). arxiv:1601.02864.
  12. 12.
    Heinlein D., Honold T., Kiermaier M., Kurz S.: Classification of binary MRD codes. (in preparation).Google Scholar
  13. 13.
    Honold T., Kiermaier M., Kurz S.: Optimal binary subspace codes of length \(6\), constant dimension \(3\) and minimum subspace distance \(4\). Contemp. Math. 632, 157–176 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Honold T., Kiermaier M., Kurz S.: Classification of large partial plane spreads in PG(6, 2) and related combinatorial objects. (2016). arxiv:1606.07655.
  15. 15.
    Honold T., Kiermaier M., Kurz S.: Constructions and bounds for mixed-dimension subspace codes. Adv. Math. Commun. 10(3), 649–682 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Honold T., Kiermaier M., Kurz S.: Partial spreads and vector space partitions. In: Greferath M., Pavčević M., Silberstein N., Vazquez-Castro A. (eds.) Network Coding and Subspace Designs. Springer, New York. (2018). arxiv:1611.06328.
  17. 17.
  18. 18.
    Johnson S.: A new upper bound for error-correcting codes. IRE Trans. Inform. Theory 8(3), 203–207 (1962).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kiermaier M., Kurz S.: An improvement of the Johnson bound for subspace codes. (2017). arxiv:1707.00650.
  20. 20.
    Kötter R., Kschischang F.R.: Coding for errors and erasures in random network coding. IEEE Trans. Inform. Theory 54(8), 3579–3591 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kurz S.: Improved upper bounds for partial spreads. Des. Codes Cryptogr. 85(1), 97–106 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kurz S.: Packing vector spaces into vector spaces. Aust. J. Comb. 68(1), 122–130 (2017).MathSciNetzbMATHGoogle Scholar
  23. 23.
    MacWilliams F.J., Sloane N.J.A.: The Theory of Error-correcting Codes. I, vol. 16. North-Holland Mathematical LibraryNorth-Holland Publishing Co, Amsterdam (1977).zbMATHGoogle Scholar
  24. 24.
    Năstase E.L., Sissokho P.A.: The maximum size of a partial spread in a finite projective space. J. Comb. Theory Ser. A 152, 353–362 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Niskanen S., Östergård P.R.J.: Cliquer user’s guide, version 1.0. Tech. Rep. T48, Communications Laboratory, Helsinki University of Technology, Espoo, Finland (2003).Google Scholar
  26. 26.
    Roth R.M.: Maximum-rank array codes and their application to crisscross error correction. IEEE Trans. Inform. Theory 37(2), 328–336 (1991).MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Xia S.T., Fu F.W.: Johnson type bounds on constant dimension codes. Des. Codes Cryptogr. 50(2), 163–172 (2009).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of BayreuthBayreuthGermany
  2. 2.ZJU-UIUC InstituteZhejiang UniversityHainingChina

Personalised recommendations