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Constructions of cyclic constant dimension codes

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Abstract

Subspace codes and particularly constant dimension codes have attracted much attention in recent years due to their applications in random network coding. As a particular subclass of subspace codes, cyclic subspace codes have additional properties that can be applied efficiently in encoding and decoding algorithms. It is desirable to find cyclic constant dimension codes such that both the code sizes and the minimum distances are as large as possible. In this paper, we explore the ideas of constructing cyclic constant dimension codes proposed in Ben-Sasson et al. (IEEE Trans Inf Theory 62(3):1157–1165, 2016) and Otal and Özbudak (Des Codes Cryptogr, doi:10.1007/s10623-016-0297-1, 2016) to obtain further results. Consequently, new code constructions are provided and several previously known results in [2] and [17] are extended.

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References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Ben-Sasson E., Etzion T., Gabizon A., Raviv N.: Subspace polynomials and cyclic subspace codes. IEEE Trans. Inf. Theory 62(3), 1157–1165 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  3. Ben-Sasson E., Kopparty S.: Affine dispersers from subspace polynomials. SIAM J. Comput. 41(4), 880–914 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  4. Ben-Sasson E., Kopparty S., Radhakrishnan J.: Subspace polynomials and limits to list decoding of Reed–Solomon codes. IEEE Trans. Inf. Theory 56(1), 113–120 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  5. Braun M., Etzion T., Ostergard P., Vardy A., Wasserman A.: Existence of \(q\)-analogues of Steiner systems. Forum Math. Pi. 4(e7), 1–14 (2016).

    MathSciNet  Google Scholar 

  6. Cheng Q., Gao S., Wan D.: Constructing high order elements through subspace polynomials. In: Proceedings of 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1463–1547 (2012).

  7. Chihara L.: On the zeros of the Askey–Wilson polynomials, with applications to coding theory. SIAM J. Math. Anal. 18(1), 191–207 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  8. Etzion T., Storme L.: Galois geometries and coding theory. Des. Codes Cryptogr. 78(1), 311–350 (2016).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Etzion T., Wachter-Zeh A.: Vector network coding based on subspace codes outperforms scalar linear network coding. In: Proceedings of IEEE International Symposium on Information Theory (ISIT), Barcelona, Spain, August 2016, pp. 1949–1953 (2016).

  11. GAP: The GAP Groups, Algorithms and Programming, Version 4.7.7. http://www.gapsystem.org (2015).

  12. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

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

  15. Lidl R., Niederreiter H.: Finite Fields. Cambridge University Press, Cambridge (2008).

    MATH  Google Scholar 

  16. Martin W.J., Zhu X.J.: Anticodes for the Grassman and bilinear forms graphs. Des. Codes Cryptogr. 6(1), 73–79 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  17. Otal K., Özbudak F.: Cyclic subspace codes via subspace polynomials. Des. Codes Cryptogr. (2016). doi:10.1007/s10623-016-0297-1.

  18. Schwartz M., Etzion T.: Codes and anticodes in the Grassman graph. J. Combin. Theory A 97(1), 27–42 (2002).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Wachter-Zeh A.: Bounds on list decoding of rank-metric codes. IEEE Trans. Inf. Theory 59(11), 7268–7277 (2013).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We sincerely thank the Associate Editor and the anonymous referees for their carefully reading and helpful suggestions which led to significant improvements of the paper. The research of Bocong Chen is supported by NSFC (Grant No. 11601158) and the Fundamental Research Funds for the Central Universities (Grant No. 2017MS111). The research of Hongwei Liu was supported by NSFC (Grant No. 11171370) and self-determined research funds of CCNU from the colleges’ basic research and operation of MOE (Grant No. CCNU14F01004).

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Authors and Affiliations

  1. School of Mathematics, South China University of Technology, Guangzhou, 510641, Guangdong, China

    Bocong Chen

  2. School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, Hubei, China

    Hongwei Liu

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  1. Bocong Chen
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  2. Hongwei Liu
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Correspondence to Hongwei Liu.

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Communicated by T. Etzion.

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Chen, B., Liu, H. Constructions of cyclic constant dimension codes. Des. Codes Cryptogr. 86, 1267–1279 (2018). https://doi.org/10.1007/s10623-017-0394-9

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