Skip to main content

On the geometry of balls in the Grassmannian and list decoding of lifted Gabidulin codes


The finite Grassmannian \(\mathcal {G}_{q}(k,n)\) is defined as the set of all \(k\)-dimensional subspaces of the ambient space \(\mathbb {F}_{q}^{n}\). Subsets of the finite Grassmannian are called constant dimension codes and have recently found an application in random network coding. In this setting codewords from \(\mathcal {G}_{q}(k,n)\) are sent through a network channel and, since errors may occur during transmission, the received words can possibly lie in \(\mathcal {G}_{q}(k',n)\), where \(k'\ne k\). In this paper, we study the balls in \(\mathcal {G}_{q}(k,n)\) with center that is not necessarily in \(\mathcal {G}_{q}(k,n)\). We describe the balls with respect to two different metrics, namely the subspace and the injection metric. Moreover, we use two different techniques for describing these balls, one is the Plücker embedding of \(\mathcal {G}_{q}(k,n)\), and the second one is a rational parametrization of the matrix representation of the codewords. With these results, we consider the problem of list decoding a certain family of constant dimension codes, called lifted Gabidulin codes. We describe a way of representing these codes by linear equations in either the matrix representation or a subset of the Plücker coordinates. The union of these equations and the linear and bilinear equations which arise from the description of the ball of a given radius provides an explicit description of the list of codewords with distance less than or equal to the given radius from the received word.

This is a preview of subscription content, access via your institution.


  1. Bossert M., Gabidulin E.M.: One family of algebraic codes for network coding. In: Proceedings of the IEEE International Symposium on Information Theory, pp. 2863–2866 (2009).

  2. Courtois N., Klimov A., Patarin J., Shamir A.: Efficient algorithms for solving overdefined systems of multivariate polynomial equations. In: Advances in cryptology—EUROCRYPT 2000 (Bruges), Lecture Notes in Computer Science, vol. 1807, pp. 392–407. Springer (2000).

  3. Delsarte P.: Bilinear forms over a finite field, with applications to coding theory. J. Comb. Theory Ser. A 25(3), 226–241 (1978).

    Google Scholar 

  4. Etzion T., Silberstein N.: Error-correcting codes in projective space via rank-metric codes and Ferrers diagrams. IEEE Trans. Inf. Theory 55(7), 2909–2919 (2009).

    Google Scholar 

  5. Etzion T., Silberstein N.: Codes and designs related to lifted MRD codes. IEEE Trans. Inf. Theory 59(2), 1004–1017 (2013).

    Google Scholar 

  6. Etzion T., Vardy A.: Error-correcting codes in projective space. IEEE Trans. Inf. Theory 57(2), 1165–1173 (2011).

    Google Scholar 

  7. Gabidulin È.M.: Theory of codes with maximum rank distance. Problemy Peredachi Informatsii 21(1), 3–16 (1985).

    Google Scholar 

  8. Gadouleau M., Yan Z.: Constant-rank codes and their connection to constant-dimension codes. IEEE Trans. Inf. Theory 56(7), 3207–3216 (2010).

    Google Scholar 

  9. Guruswami V., Wang C.: Explicit rank-metric codes list-decodable with optimal redundancy, arXiv:1311.7084 [cs.IT] (2013).

  10. Guruswami V., Xing C.: List decoding Reed–Solomon, algebraic–geometric, and Gabidulin subcodes up to the singleton bound, electronic collloquium on computationl complexity. Report No. 146 (2012).

  11. Guruswami V., Narayanan S., Wang C.: List decoding subspace codes from insertions and deletions. In: Proceedings of Innovations in Theoretical Computer Science (ITCS 2012), pp. 183–189 (2012).

  12. Hodge W.V.D., Pedoe D.: Methods of Algebraic Geometry, vol. 2. Cambridge University Press, Cambridge (1952).

  13. Kipnis A., Shamir A.: Cryptanalysis of the HFE public key cryptosystem. In: Advances in Cryptology—CRYPTO’99, Santa Barbara. Lecture Notes in Computer Science, vol. 1666, pp. 19–30. Springer: Berlin (1999).

  14. Kleiman S.L., Laksov D.: Schubert calculus. Am. Math. Mon. 79, 1061–1082 (1972).

    Google Scholar 

  15. Kohnert A., Kurz S.: Construction of large constant-dimension codes with a prescribed minimum distance. Lecture Notes in Computer Science, vol. 5393, pp. 31–42. Springer: Berlin (2008).

  16. Kötter R., Kschischang F.R.: Coding for errors and erasures in random network coding. IEEE Trans. Inf. Theory 54(8), 3579–3591 (2008).

    Google Scholar 

  17. Manganiello F.. Gorla E., Rosenthal J.: Spread codes and spread decoding in network coding. In: Proceedings of International Symposium on Information Theory, pp. 881–885, Toronto, ON, Canada (2008).

  18. Mahdavifar H., Vardy A.: Algebraic list-decoding on the operator channel. In: Proceedings of the IEEE International Symposium on Information Theory (ISIT), pp. 1193–1197 (2010).

  19. Mahdavifar H., Vardy A.: List-decoding of subspace codes and rank-metric codes up to Singleton bound. In: Proceedings of the IEEE International Symposium on Information Theory (ISIT), pp. 1483–1492 (2012).

  20. Procesi C.: A primer of invariant theory. Brandeis Lecture Notes, Brandeis University, 1982, Notes by G. Boffi

  21. Rosenthal J., Trautmann A.-L.: Decoding of subspace codes, a problem of schubert calculus over finite fields. Mathematical System Theory—Festschrift in Honor of Uwe Helmke on the Occasion of his Sixtieth Birthday, CreateSpace (2012).

  22. Roth R.M.: Maximum-rank array codes and their application to crisscross error correction. IEEE Trans. Inf. Theory, 37(2), 328–336 (1991).

    Google Scholar 

  23. Silberstein N., Etzion T.: Enumerative coding for Grassmannian space. IEEE Trans. Inf. Theory, 57(1), 365–374 (2011).

    Google Scholar 

  24. Silberstein N., Etzion T.: Large constant dimension codes and lexicodes. Adv. Math. Commun. 5(2), 177–189 (2011).

    Google Scholar 

  25. 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).

    Google Scholar 

  26. Skachek V.: Recursive code construction for random networks. IEEE Trans. Inf. Theory 56(3), 1378–1382 (2010).

    Google Scholar 

  27. Thomae E., Wolf C.: Solving systems of multivariate quadratic equations over finite fields or: from relinearization to MutantXL, Cryptology ePrint Archive, Report 2010/596, 2010,

  28. Trautmann A.-L., Manganiello F., Braun M., Rosenthal J.: Cyclic orbit codes. IEEE Trans. Inf. Theory 59(11), 7386–7404 (2013).

    Google Scholar 

  29. Trautmann A.-L.: Plücker embedding of cyclic orbit codes. In: Proceedings of the 20th International Symposium on Mathematical Theory of Networks and Systems—MTNS, Melbourne, pp. 1–15(2012).

  30. Trautmann A.-L., Silberstein N., Rosenthal J.: List decoding of lifted Gabidulin codes via the Plücker embedding. In: Preproceedings of the International Workshop on Coding and Cryptography (WCC), Bergen, Norway, pp. 539–549 (2013).

  31. Trautmann A.L.: Constructions, decoding and automorphisms of subspace codes. PhD thesis, University of Zurich, Switzerland (2013).

  32. Trautmann A.-L., Manganiello F., Rosenthal J.: Orbit codes—a new concept in the area of network coding. In: Proceedings of IEEE Information Theory Workshop (ITW), Dublin, Ireland, pp. 1–4 (2010).

  33. Trautmann A.-L., Rosenthal J.: New improvements on the Echelon–Ferrers construction. In: Proceedings of the International Symposium on Mathematical Theory of Networks and Systems, pp. 405–408 (2010).

  34. Wachter-Zeh A.: Bounds on list decoding Gabidulin codes. IEEE Trans. Inf. Theory, pp. 7268–7277 (2013).

  35. Wachter-Zeh A., Zeh A.: Interpolation-based decoding of interleaved Gabidulin codes. In: Preproceedings of the International Workshop on Coding and Cryptography (WCC), Bergen, Norway, pp. 528–538 (2013).

Download references


The authors wish to thank Antonia Wachter-Zeh for many helpful discussions. They also thank the anonymous reviewers for their valuable comments and suggestions that helped to improve the presentation of the paper. J. Rosenthal was partially supported by Swiss National Science Foundation Grant Nos. 138080 and 149716. N. Silberstein was supported in part at the Technion by a Fine Fellowship. A.-L. Trautmann was partially supported by Forschungskredit of the University of Zurich, Grant No. 57104103, and Swiss National Science Foundation Fellowship No. 147304. Parts of this work were presented at the International Workshop on Coding and Cryptography 2013 in Bergen, Norway, and appear in its proceedings [30].

Author information

Authors and Affiliations


Corresponding author

Correspondence to Anna-Lena Trautmann.

Additional information

This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding and Cryptography”.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Rosenthal, J., Silberstein, N. & Trautmann, AL. On the geometry of balls in the Grassmannian and list decoding of lifted Gabidulin codes. Des. Codes Cryptogr. 73, 393–416 (2014).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Grassmannian
  • Projective space
  • Subspace codes
  • Network coding
  • List decoding

Mathematics Subject Classification

  • 11T71
  • 14G50