Skip to main content
SpringerLink
Log in

A Construction for Constant Dimension Codes from the Known Codes

  • Conference paper
  • First Online:
Wireless Algorithms, Systems, and Applications (WASA 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12937))

Abstract

One of the most fundamental topics in subspace coding is to explore the maximal possible value \(\mathbf{A}_q(n,d,k)\) of a set of k-dimensional subspaces in \(\mathbb F_q^n\) such that the subspace distance satisfies \(\text {d}_{\text {S}}(U,V) = \dim (U+V)-\dim (U\cap V) \ge d\) for any two different k-dimensional subspaces U and V in this set. In this paper, we propose a construction for constant dimension subspace codes from the existing results. This construction is done by merging two existing constructions, which exceeds the latest improvements including the cases: \(A_2(8,4,3)\ge 1331\) and \(A_2(8,4,4)\ge 4802\).

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

References

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

    Google Scholar 

  2. Braun, M.J., Ostergrad, P.R., Wassermann, A.: New lower bounds for binary constant-dimension subspace codes. Exp. Math. 27(2), 179–183 (2018). https://doi.org/10.1080/10586458.2016.1239145

    Article  MathSciNet  MATH  Google Scholar 

  3. Braun, M., Kerber, A., Laue, R.: Systematic construction OFQ-analogs of (v, k,)-designs. Des. Codes Crypt. 34(1), 55–70 (2005)

    Article  Google Scholar 

  4. Braun, M., Ostergard, P.R.J., Wassermann, A.: New lower bounds for binary constant-dimension subspace codes. Exp. Math. 27(2), 179–183 (2016). https://doi.org/10.1080/10586458.2016.1239145

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, H., He, X., Weng, J., Xu, L.: New constructions of subspace codes using subsets of MRD codes in several blocks. IEEE Trans. Inf. Theory 66(9), 5317–5321 (2020). http://arxiv.org/abs/1908.03804

  6. Cossidente, A., Kurz, S., Marino, G., Pavese, F.: Combining subspace codes. CoRR abs/1911.03387 (2019)

    Google Scholar 

  7. Cruz, J.D.L., Gorla, E., Lopez, H.H., Ravagnani, A.: Rank distribution of Delsarte codes. Mathematics (2015)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. Diestel, R.: Graph theory. Math. Gazette 173(502), 67–128 (2000)

    MATH  Google Scholar 

  10. Duo, L.: Discrete Mathematics and Applications. Tsinghua University Press (2013). (in Chinese)

    Google Scholar 

  11. He, X.: A hierarchical-based greedy algorithm for Echelon-Ferrers construction. arXiv (2019)

    Google Scholar 

  12. 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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  14. Gutirrez-Garca, I., Naizir, I.M.: Finding cliques in projective space: a method for construction of cyclic Grassmannian codes. IEEE Access 8, 51333–51339 (2020). https://doi.org/10.1109/ACCESS.2020.2980670

    Article  Google Scholar 

  15. He, X., Chen, Y., Zhang, Z.: Improving the linkage construction with Echelon-Ferrers for constant-dimension codes. IEEE Commun. Lett. 24, 1875–1879 (2020)

    Article  Google Scholar 

  16. He, X.: Construction of constant dimension code from two parallel versions of linkage construction. IEEE Commun. Lett. 24(11), 2392–2395 (2020)

    Article  Google Scholar 

  17. Heinlein, D.: Generalized linkage construction for constant-dimension codes. arxiv.org (2019)

    Google Scholar 

  18. Heinlein, D., Kiermaier, M., Kurz, S., Wassermann, A.: Tables of subspace codes. arxiv.org (2016)

    Google Scholar 

  19. Heinlein, D., Kurz, S.: COSET construction for subspace codes. IEEE Trans. Inf. Theory PP(99), 1 (2015)

    Google Scholar 

  20. Heinlein, D., Kurz, S.: Asymptotic bounds for the sizes of constant dimension codes and an improved lower bound. In: Coding Theory and Applications, ICMCTA 2017 (2017)

    Google Scholar 

  21. Kuhn, H.W.: The Hungarian method for the assignment problem. Naval Res. Logist. Quart. 2, 83–97 (1955)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  23. 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). https://doi.org/10.1007/978-3-540-89994-5_4

    Chapter  Google Scholar 

  24. Kurz, S.: A note on the linkage construction for constant dimension codes. arXiv:1906.09780, https://arxiv.org/abs/1906.09780 (2019)

  25. Kurz, S.: Lifted codes and the multilevel construction for constant dimension codes. arXiv:2004.14241 (2020)

  26. Lao, H., Chen, H., Weng, J., Tan, X.: Parameter-controlled inserting constructions of constant dimension subspace codes. arxiv.org (2020)

    Google Scholar 

  27. Li, F.: Construction of constant dimension subspace codes by modifying linkage construction. IEEE Trans. Inf. Theory 66(5), 2760–2764 (2020)

    Article  MathSciNet  Google Scholar 

  28. Li, S., Yeung, R., Cai, N.: Linear network coding. IEEE Trans. Inf. Theory 49(2), 371–381 (2003)

    Article  MathSciNet  Google Scholar 

  29. Munkres, J.: Algorithms for the assignment and transportation problems. SIAM. J 10, 196–210 (1962)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  31. 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)

    Article  MathSciNet  Google Scholar 

  32. Xu, L., Chen, H.: New constant-dimension subspace codes from maximum rank-distance codes. IEEE Trans. Inf. Theory 64(9), 6315–6319 (2018)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgement

The work is supported by the Science and Technology Planning Project of Guangdong Province (No. 190827105555406, 2019B010116001), the Natural Science Foundation of Guangdong Province (No. 2020A1515010899), the Key Scientific Research Project of Universities in Guangdong Province (No. 2020ZDZX3028), the innovation strong school project of Guangdong Province (No. 2020K2D2X1201) and the Natural Science Foundation of China (NO.61672303,61872083,61872081).

Author information

Authors and Affiliations

  1. Dongguan University of Technology, Dongguan, China

    Kunxiao Zhou, Zusheng Zhang, Feng Shi & Xianmang He

  2. Shantou University, Shantou, China

    Yindong Chen

Authors
  1. Kunxiao Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yindong Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zusheng Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Feng Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Xianmang He
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xianmang He .

Editor information

Editors and Affiliations

  1. Nanjing University of Aeronautics and Astronautics, Nanjing, China

    Zhe Liu

  2. Shanghai Jiao Tong University, Shanghai, China

    Fan Wu

  3. Missouri University of Science and Technology, Rolla, MO, USA

    Sajal K. Das

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Zhou, K., Chen, Y., Zhang, Z., Shi, F., He, X. (2021). A Construction for Constant Dimension Codes from the Known Codes. In: Liu, Z., Wu, F., Das, S.K. (eds) Wireless Algorithms, Systems, and Applications. WASA 2021. Lecture Notes in Computer Science(), vol 12937. Springer, Cham. https://doi.org/10.1007/978-3-030-85928-2_20

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-85928-2_20

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-85927-5

  • Online ISBN: 978-3-030-85928-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation