Designs, Codes and Cryptography

, Volume 87, Issue 1, pp 107–121 | Cite as

Constructions of optimal Ferrers diagram rank metric codes

  • Tao ZhangEmail author
  • Gennian Ge


Subspace codes and constant dimension codes have become a widely investigated research topic due to their significance to error control in random linear network coding. Rank metric codes in Ferrers diagrams can be used to construct good subspace codes and constant dimension codes. In this paper, three constructions of Ferrers diagram rank metric codes are presented. The first two constructions are based on subcodes of maximum rank distance codes, and the last one generates new codes from known Ferrers diagram rank metric codes. Each of these constructions produces optimal codes with different diagrams and parameters for which no optimal construction was known before.


Ferrers diagram Rank metric code Gabidulin code Subspace code Constant dimension code 

Mathematics Subject Classification

15A03 15A99 15B99 



The authors express their gratitude to the anonymous reviewers for their detailed and constructive comments which are very helpful to the improvement of this paper, and to Prof. Tuvi Etzion, the Associate Editor, for his insightful advice and excellent editorial job.


  1. 1.
    Bachoc C., Passuello A., Vallentin F.: Bounds for projective codes from semidefinite programming. Adv. Math. Commun. 7(2), 127–145 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    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
  3. 3.
    Etzion T., Gorla E., Ravagnani A., Wachter-Zeh A.: Optimal Ferrers diagram rank-metric codes. IEEE Trans. Inf. Theory 62(4), 1616–1630 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Etzion T., Silberstein N.: Error-correcting codes in projective spaces via rank-metric codes and Ferrers diagrams. IEEE Trans. Inf. Theory 55(7), 2909–2919 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Etzion T., Silberstein N.: Codes and designs related to lifted MRD codes. IEEE Trans. Inf. Theory 59(2), 1004–1017 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Etzion T., Vardy A.: Error-correcting codes in projective space. IEEE Trans. Inf. Theory 57(2), 1165–1173 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gabidulin E.M.: Theory of codes with maximum rank distance. Problemy Peredachi Informatsii 21(1), 3–16 (1985).MathSciNetzbMATHGoogle Scholar
  8. 8.
    Gadouleau M., Yan Z.: Constant-rank codes and their connection to constant-dimension codes. IEEE Trans. Inf. Theory 56(7), 3207–3216 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Koetter R., Kschischang F.R.: Coding for errors and erasures in random network coding. IEEE Trans. Inf. Theory 54(8), 3579–3591 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 9.
    Kohnert A., Kurz S.: Construction of large constant dimension codes with a prescribed minimum distance. In: Mathematical Methods in Computer Science. Lecture Notes in Computer Science, vol 5393, pp. 31–42. Springer, Berlin (2008).Google Scholar
  11. 10.
    Silberstein N., Etzion T.: Enumerative coding for Grassmannian space. IEEE Trans. Inf. Theory 57(1), 365–374 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 11.
    Silberstein N., Etzion T.: Large constant dimension codes and lexicodes. Adv. Math. Commun. 5(2), 177–189 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 12.
    Silberstein N., Trautmann A.-L.: Subspace codes based on graph matchings, Ferrers diagrams, and pending blocks. IEEE Trans. Inf. Theory 61(7), 3937–3953 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 13.
    Silberstein N., Trautmann A.-L.: New lower bounds for constant dimension codes. In: Proceedings of the IEEE International Symposium on Information Theory, pp. 514–518 (2013).Google Scholar
  15. 14.
    Silva D., Kschischang F.R.: On metrics for error correction in network coding. IEEE Trans. Inf. Theory 55(12), 5479–5490 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 15.
    Silva D., Kschischang F.R., Kötter R.: A rank-metric approach to error control in random network coding. IEEE Trans. Inf. Theory 54(9), 3951–3967 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 16.
    Skachek V.: Recursive code construction for random networks. IEEE Trans. Inf. Theory 56(3), 1378–1382 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 17.
    Trautmann A.-L., Rosenthal J.: New improvements on the Echelon-Ferrers construction. In: Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems, pp. 405–408 (2010).Google Scholar
  19. 18.
    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.School of Mathematics and Information ScienceGuangzhou UniversityGuangzhouChina

Personalised recommendations