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Subspaces intersecting in at most a point

  • Sascha KurzEmail author
Article
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Abstract

We improve on the lower bound of the maximum number of planes in \({\text {PG}}(8,q)\cong \mathbb {F}_q^{9}\) pairwise intersecting in at most a point. In terms of constant dimension codes this leads to \(A_q(9,4;3)\ge q^{12}+ 2q^8+2q^7+q^6+2q^5+2q^4-2q^2-2q+1\). This result is obtained via a more general construction strategy, which also yields other improvements.

Keywords

Constant dimension codes Finite projective geometry Network coding 

Mathematics Subject Classification

Primary 51E20 Secondary 05B25 94B65 

Notes

Acknowledgements

The author would like to thank Thomas Honold for his analysis of possible clique sizes in the constant dimension codes from [9, Lemma 12, Example 4] and [8, Theorem 4], see Footnote 3. The main idea for Theorem 3 is inspired by [2]. Further thanks go to the anonymous referees for their careful reading and helpful remarks.

References

  1. 1.
    Ai J., Honold T., Liu H.: The expurgation-augmentation method for constructing good plane subspace codes. arXiv:1601.01502 (2016).
  2. 2.
    Cossidente A., Marino G., Pavese F.: Subspace code constructions. arXiv:1905.11021 (2019).
  3. 3.
    Cossidente A., Pavese F.: On subspace codes. Des. Codes Cryptogr. 78(2), 527–531 (2016).MathSciNetCrossRefGoogle Scholar
  4. 4.
    Delsarte P.: Bilinear forms over a finite field, with applications to coding theory. J. Comb. Theory Ser. A 25(3), 226–241 (1978).MathSciNetCrossRefGoogle Scholar
  5. 5.
    Etzion T., Silberstein N.: Codes and designs related to lifted MRD codes. IEEE Trans. Inform. Theory 59(2), 1004–1017 (2013).MathSciNetCrossRefGoogle Scholar
  6. 6.
    Heinlein D., Honold T., Kiermaier M., Kurz S., Wassermann A.: Classifying optimal binary subspace codes of length 8, constant dimension 4 and minimum distance 6. Des. Codes Cryptogr. 87(2–3), 375–391 (2019).MathSciNetCrossRefGoogle Scholar
  7. 7.
    Heinlein D., Kiermaier M., Kurz S., Wassermann A.: Tables of subspace codes. arXiv:1601.02864 (2016).
  8. 8.
    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).CrossRefGoogle Scholar
  9. 9.
    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, Volume 632 of Contemp. Math., pp. 157–176. Amer. Math. Soc., Providence, RI (2015).Google Scholar
  10. 10.
    Honold T., Kiermaier M., Kurz S.: Partial spreads and vector space partitions. In Network Coding and Subspace Designs, Signals and Communication Technology, pp. 131–170. Springer, Cham (2018).CrossRefGoogle Scholar
  11. 11.
    Kötter R., Kschischang F.R.: Coding for errors and erasures in random network coding. IEEE Trans. Inform. Theory 54(8), 3579–3591 (2008).MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kurz S.: A note on the linkage construction for constant dimension codes. arXiv:1906.09780 (2019).
  13. 13.
    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).MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.University of BayreuthBayreuthGermany

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