Skip to main content
SpringerLink
Log in

Asymptotic Bounds for the Sizes of Constant Dimension Codes and an Improved Lower Bound

  • Conference paper
  • First Online:
Coding Theory and Applications (ICMCTA 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10495))

Included in the following conference series:

Abstract

We study asymptotic lower and upper bounds for the sizes of constant dimension codes with respect to the subspace or injection distance, which is used in random linear network coding. In this context we review known upper bounds and show relations between them. A slightly improved version of the so-called linkage construction is presented which is e.g. used to construct constant dimension codes with subspace distance \(d=4\), dimension \(k=3\) of the codewords for all field sizes q, and sufficiently large dimensions v of the ambient space. It exceeds the MRD bound, for codes containing a lifted MRD code, by Etzion and Silberstein.

The work was supported by the ICT COST Action IC1104 and grants KU 2430/3-1, WA 1666/9-1 – “Integer Linear Programming Models for Subspace Codes and Finite Geometry” – from the German Research Foundation.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    By a tedious computation one can check that the sphere-packing bound is strictly tighter than the Singleton bound iff \(q=2\), \(v=2k\) and \(d=6\).

  2. 2.

    It can be verified that for \(2k \le v \le 3k-1\) the optimal choice of \(\varDelta \) in [38, Corollary39] is given by \(\varDelta =v-k\). In that case the construction is essentially the union of a lifted MRD code with an \((v-k,\#\mathcal {C}',d;k)_q\) code \(\mathcal {C}'\). Note that for \(v-k < \varDelta \le v\) the constructed code is an embedded \((\varDelta ,\#\mathcal {C}',d;k)_q\) code \(\mathcal {C}'\).

  3. 3.

    Entries of type improved_linkage(m) correspond to Corollary 4 with m chosen as parameter.

References

  1. Agrell, E., Vardy, A., Zeger, K.: Upper bounds for constant-weight codes. IEEE Trans. Inform. Theory 46(7), 2373–2395 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ahlswede, R., Aydinian, H.: On error control codes for random network coding. In: Workshop on Network Coding, Theory, and Applications, NetCod 2009, pp. 68–73. IEEE (2009)

    Google Scholar 

  3. Ahlswede, R., Aydinian, H.K., Khachatrian, L.H.: On perfect codes and related concepts. Des. Codes Crypt. 22(3), 221–237 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ahlswede, R., Cai, N., Li, S.Y.R., Yeung, R.W.: Network information flow. IEEE Trans. Inf. Theory 46(4), 1204–1216 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ai, J., Honold, T., Liu, H.: The expurgation-augmentation method for constructing good plane subspace codes. arXiv preprint arXiv:1601.01502 (2016)

  6. Bachoc, C., Passuello, A., Vallentin, F.: Bounds for projective codes from semidefinite programming. Adv. Math. Commun. 7(2), 127–145 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Beutelspacher, A.: Partial spreads in finite projective spaces and partial designs. Math. Z. 145(3), 211–229 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  8. Blackburn, S.R., Etzion, T.: The asymptotic behavior of grassmannian codes. IEEE Trans. Inform. Theory 58(10), 6605–6609 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Braun, M., Etzion, T., Östergård, P.R.J., Vardy, A., Wassermann, A.: Existence of \(q\)-analogs of steiner systems. Forum Math. Pi 4, 1–14 (2016)

    Google Scholar 

  10. Braun, M., Östergård, P.R.J., Wassermann, A.: New lower bounds for binary constant-dimension subspace codes. Exp. Math., 1–5. doi:10.1080/10586458.2016.1239145

  11. Delsarte, P.: Hahn polynomials, discrete harmonics, and \(t\)-designs. SIAM J. Appl. Math. 34(1), 157–166 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  12. Delsarte, P.: An algebraic approach to the association schemes of coding theory. Ph.D. thesis, Philips Research Laboratories (1973)

    Google Scholar 

  13. Drake, D., Freeman, J.: Partial \(t\)-spreads and group constructible \((s, r,\mu )\)-nets. J. Geom. 13(2), 210–216 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  14. El-Zanati, S., Jordon, H., Seelinger, G., Sissokho, P., Spence, L.: The maximum size of a partial \(3\)-spread in a finite vector space over \({G}{F}(2)\). Des. Codes Crypt. 54(2), 101–107 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. 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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. Frankl, P., Rödl, V.: Near perfect coverings in graphs and hypergraphs. Eur. J. Comb. 6(4), 317–326 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  19. Frankl, P., Wilson, R.M.: The Erdős-Ko-Rado theorem for vector spaces. J. Comb. Theory, Ser. A 43(2), 228–236 (1986)

    Article  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  21. Gluesing-Luerssen, H., Morrison, K., Troha, C.: Cyclic orbit codes and stabilizer subfields. Adv. Math. Commun. 9(2), 177–197 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Gluesing-Luerssen, H., Troha, C.: Construction of subspace codes through linkage. Adv. Math. Commun. 10(3), 525–540 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  23. Heinlein, D., Kiermaier, M., Kurz, S., Wassermann, A.: Tables of subspace codes. arXiv preprint arXiv:601.02864 (2016)

  24. Heinlein, D., Kurz, S.: A new upper bound for subspace codes. arXiv preprint arXiv:1703.08712 (2017)

  25. Honold, T., Kiermaier, M.: On putative \(q\)-analogues of the Fano plane and related combinatorial structures. In: Dynamical Systems, Number Theory and Applications, pp. 141–175. World Sci. Publ, Hackensack, NJ (2016)

    Google Scholar 

  26. Honold, T., Kiermaier, M., Kurz, S.: Optimal binary subspace codes of length \(6\), constant dimension \(3\) and minimum subspace distance \(4\). In: Topics in finite fields, Contemp. Math., v ol. 632, pp. 157–176. Amer. Math. Soc., Providence, RI (2015)

    Google Scholar 

  27. Honold, T., Kiermaier, M., Kurz, S.: Partial spreads and vector space partitions. arXiv preprint arXiv:1611.06328 (2016)

  28. Johnson, S.: A new upper bound for error-correcting codes. IRE Trans. Inform. Theory 8(3), 203–207 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  29. Khaleghi, A., Silva, D., Kschischang, F.R.: Subspace codes. In: Parker, M.G. (ed.) IMACC 2009. LNCS, vol. 5921, pp. 1–21. Springer, Heidelberg (2009). doi:10.1007/978-3-642-10868-6_1

    Chapter  Google Scholar 

  30. Kiermaier, M., Kurz, S., Wassermann, A.: The order of the automorphism group of a binary \(q\)-analog of the fano plane is at most two. Des. Codes Crypt. (2017). doi:10.1007/s10623-017-0360-6

  31. Kohnert, A., Kurz, S.: Construction of large constant dimension codes with a prescribed minimum distance. In: Calmet, J., Geiselmann, W., Müller-Quade, J. (eds.) Mathematical Methods in Computer Science. LNCS, vol. 5393, pp. 31–42. Springer, Heidelberg (2008). doi:10.1007/978-3-540-89994-5_4

    Chapter  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  33. Kurz, S.: Improved upper bounds for partial spreads. Des. Codes Crypt. 85(1), 97–106 (2017). doi:10.1007/s10623-016-0290-8

  34. Kurz, S.: Packing vector spaces into vector spaces. Australas. J. Comb. 68(1), 122–130 (2017)

    MathSciNet  Google Scholar 

  35. MacWilliams, F.J., Sloane, N.J.A.: The theory of error-correcting codes. II. North-Holland Publishing Co., Amsterdam-New York-Oxford , North-Holland Mathematical Library, vol. 16 (1977)

    Google Scholar 

  36. Năstase, E., Sissokho, P.: The maximum size of a partial spread in a finite projective space. arXiv preprint arXiv:1605.04824 (2016)

  37. Segre, B.: Teoria di galois, fibrazioni proiettive e geometrie non desarguesiane. Annali di Matematica 64(1), 1–76 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  38. Silberstein, N., Trautmann, A.L.: Subspace codes based on graph matchings, ferrers diagrams, and pending blocks. IEEE Trans. Inform. Theory 61(7), 3937–3953 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  39. Silva, D., Kschischang, F., Kötter, R.: A rank-metric approach to error control in random network coding. IEEE Trans. Inform. Theory 54(9), 3951–3967 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  40. Silva, D., Kschischang, F.R.: On metrics for error correction in network coding. IEEE Trans. Inform. Theory 55(12), 5479–5490 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  41. Tonchev, V.D.: Codes and designs. In: Handbook of Coding Theory, vol. 2, pp. 1229–1267 (1998)

    Google Scholar 

  42. Wang, H., Xing, C., Safavi-Naini, R.: Linear authentication codes: bounds and constructions. IEEE Trans. Inform. Theory 49(4), 866–872 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  43. Xia, S.T., Fu, F.W.: Johnson type bounds on constant dimension codes. Des. Codes Crypt. 50(2), 163–172 (2009)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgement

The authors would like to thank Harout Aydinian for providing an enlarged proof of Theorem 8, Natalia Silberstein for explaining the restriction \(3k \le v\) in [38, Corollary 39], Heide Gluesing-Luerssen for clarifying the independent origin of the linkage construction, and Alfred Wassermann for discussions about the asymptotic results of Frankl and Rödl. We thank the reviewers for their comments that helped us to improve the presentation of the paper.

Author information

Authors and Affiliations

  1. University of Bayreuth, Bayreuth, Germany

    Daniel Heinlein & Sascha Kurz

Authors
  1. Daniel Heinlein
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sascha Kurz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daniel Heinlein .

Editor information

Editors and Affiliations

  1. University of Valladolid, Valladolid, Spain

    Ángela I. Barbero

  2. University of Tartu, Tartu, Estonia

    Vitaly Skachek

  3. Simula@UiB and University of Bergen, Bergen, Norway

    Øyvind Ytrehus

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Heinlein, D., Kurz, S. (2017). Asymptotic Bounds for the Sizes of Constant Dimension Codes and an Improved Lower Bound. In: Barbero, Á., Skachek, V., Ytrehus, Ø. (eds) Coding Theory and Applications. ICMCTA 2017. Lecture Notes in Computer Science(), vol 10495. Springer, Cham. https://doi.org/10.1007/978-3-319-66278-7_15

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-66278-7_15

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-66277-0

  • Online ISBN: 978-3-319-66278-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation