Abstract
Let V be a k-dimensional subspace of \({{\mathbb {F}}}_{q^n}\). Define \(a{{\mathbb {F}}}_q=\{a\lambda :\lambda \in {{\mathbb {F}}}_q\}\). We call V the Sidon space if any nonzero a, b, c and \(d\in V\) such that \(ab=cd\), then \(\{a{{\mathbb {F}}}_q,b{{\mathbb {F}}}_q\}=\{c{{\mathbb {F}}}_q,d{{\mathbb {F}}}_q\}\). We first provide the new results for high-dimensional Sidon spaces by using the direct sum of two small-dimensional Sidon spaces. Besides, we develop and generalize the constructions of cyclic subspace codes presented in Niu (Discret Math 343(5):111788, 2020) and Feng (Discret Math 344(4):112273, 2021), and further obtain several large ones without changing minimum distance through combining the orbits of distinct Sidon spaces.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahlswede R., Aydinian H.K., Khachatrian L.H.: On perfect codes and related concepts. Des. Codes Cryptogr. 22(3), 221–237 (2001).
Bachoc C., Serra O.: Z\(\rm \acute{e}\)mor: an analogue of Vosper’s theorem for extension fields. Math. Proc. Cambridge Philos. Soc. 163(3), 423–452 (2017).
Ben-Sasson E., Etzion T., Gabizon A., Raviv N.: Subspace polynomials and cyclic subspace codes. IEEE Trans. Inf. Theory 62(3), 1157–1165 (2016).
Braun M., Etzion T., Ostergard P., Vardy A., Wasserman A.: Existence of q-analogs of Steiner systems. Forum Math. Pi. 4(e7), 1–14 (2016).
Chen B., Liu H.: Constructions of cyclic constant dimension codes. Des. Codes Cryptogr. 1–13 (2017). https://doi.org/10.1007/s10623-017-0394-9.
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).
Chihara L.: On the zeros of the Askey-Wilson polynomials, with applications to coding theory. SIAM J. Math. Anal. 18(1), 191–207 (1987).
Erdős P.: Solved and unsolved problems in combinatorics and combinatorial number theory. Congr. Numer. 4, 49–62 (1981).
Etzion T., Vardy A.: Error-correcting codes in projective space. IEEE Trans. Inf. Theory 57(2), 1165–1173 (2011).
Feng T., Wang Y.: New constructions of large cyclic subspace codes and Sidon spaces. Discret. Math. 344(4), 112273 (2021).
Gluesing-Luerssen H., Lehmann H.: Distance distributions of cyclic orbit codes. Des. Codes Cryptogr. 89, 447–470 (2021).
Gluesing-Luerssen H., Morrison K., Troha C.: Cyclic orbit codes and stabilizer subfields. Adv. Math. Commun. 9(2), 177–197 (2015).
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).
Kötter R., Kschischang F.R.: Coding for errors and erasures in random network coding. IEEE Trans. Inf. Theory 54(8), 3579–3591 (2008).
Li Y., Liu H.: Cyclic subspace codes via the sum of Sidon spaces arXiv:2105.12520 [cs.IT].
Niu Y., Yue Y., Wu Y.: Several kinds of large cyclic subspace codes via Sidon spaces. Discret. Math. 343(5), 111788 (2020).
Otal K., Özbudak F.: Cyclic subspace codes via subspace polynomials. Des. Codes Cryptogr. 85(2), 191–204 (2017).
Roth R.M., Raviv N., Tamo I.: Construction of Sidon spaces with applications to coding. IEEE Trans. Inf. Theory 64(6), 4412–4422 (2018).
Schwartz M., Etzion T.: Codes and anticodes in the Grassman graph. J. Combin. Theory A 97(1), 27–42 (2002).
Trautmann A.L., Manganiello F., Braun M., Rosenthal J.: Cyclic orbit codes. IEEE Trans. Inf. Theory 59(11), 7386–7404 (2013).
Zhang H., Cao X.: Further constructions of cyclic subspace codes. Cryptogr. Commun. 13, 245–262 (2021).
Zhao W., Tang X.: A characterization of cyclic subspace codes via subspace polynomials. Finite Fields Appl. 57, 1–12 (2019).
Acknowledgements
We are grateful to the anonymous referees for useful comments and suggestions. Particular thanks to Professor Junwu Dong for many helpful discussions and to Yuting Liu for correcting grammar errors in this paper.
Author information
Authors and Affiliations
Corresponding authors
Additional information
Communicated by G. Lunardon.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work were supported by the National Natural Science Foundation of China (Grant Nos. 61772147 and 12171114) and the Innovation Research for the Postgraduates of Guangzhou University (Grant No. 2021GDJC-D07)
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Zhang, H., Tang, C. Constructions of large cyclic constant dimension codes via Sidon spaces. Des. Codes Cryptogr. 91, 29–44 (2023). https://doi.org/10.1007/s10623-022-01095-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-022-01095-3