Abstract
Subspace codes are of great use in noncoherent linear network coding (Katti et al in ACM SIGCOMM Comput Commun Rev 38(4):401–412, 2008; Kötter and Kschischang in IEEE Trans Inform Theory 54(8):3579–3591, 2008; Silva et al. in IEEE Trans Inform Theory 54(9):3951–3967, 2008). As a particular subclass of subspace codes, cyclic constant dimension subspace codes can be encoded and decoded more efficiently. There is an increased interest in constructing cyclic constant dimension subspace codes whose sizes and minimum distances are as large as possible. Roth et al. (IEEE Trans Inform Theory 64(6):4412–4422, 2017) constructed several cyclic constant dimension subspace codes using Sidon spaces. In this paper, we present a criterion which can be used to determine whether or not the sum of some distinct Sidon spaces is again a Sidon space. Based on this result, we obtain cyclic constant dimension subspace codes via the sum of several Sidon spaces. Our results generalize some results in Niu et al. (Discret Math 343(5):111788, 2020) and Roth et al. (IEEE Trans Inform Theory 64(6):4412–4422, 2017).
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References
Bachoc C., Serra O., Zemor G.: An analogue of Vosper’s theorem for extension fields. Math. Proc. Camb. Philos. Soc. 163(3), 423–452 (2017).
Ben-Sasson E., Kopparty S., Radhakrishnan J.: Subspace polynomials and limits to list decoding of Reed–Solomon codes. IEEE Trans. Inform. Theory 56(1), 113–120 (2010).
Ben-Sasson E., Etzion T., Gabizon A., Raviv N.: Subspace polynomials and cyclic subspace codes. IEEE Trans. Inform. Theory 62(3), 1157–1165 (2016).
Chen B., Liu H.: Constructions of cyclic constant dimension codes. Des. Codes Cryptogr. 86(6), 1267–1279 (2017).
Etzion T., Vardy A.: Error-correcting codes in projective space. IEEE Trans. Inform. 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).
Gluesing-Luerssen H., Lehmann H.: Automorphism groups and isometries for cyclic orbit codes. arXiv:2101.09548
Katti S., Katabi D., Balakrishnan H., Medard M.: Symbol-level network coding for wireless mesh networks. ACM SIGCOMM Comput. Commun. Rev. 38(4), 401–412 (2008).
Kohnert A., Kurz S.: Constructing of large constant dimension codes with a prescribed minimum distance. Math. Methods Comput. Sci. 5393, 31–42 (2008).
Kötter R., Kschischang F.R.: Coding for errors and erasures in random network coding. IEEE Trans. Inform. Theory 54(8), 3579–3591 (2008).
Lidl R., Niederreiter H.: Finite Fields. Cambridge University Press, Cambridge (1997).
Niu Y., Yue Q., 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).
Rosenthal J., Trautmann A.L.: A complete characterization of irreducible cyclic orbit codes and their Plücker embedding. Des. Codes Cryptogr. 66, 275–289 (2013).
Roth R., Raviv N., Tamo I.: Construction of Sidon spaces with applications to coding. IEEE Trans. Inform. Theory 64(6), 4412–4422 (2017).
Silva D., Kschischang F.R., Kötter R.: A rank-metric approach to error control in random network coding. IEEE Trans. Inform. Theory 54(9), 3951–3967 (2008).
Trautmann A.L.: Isometry and automorphisms of constant dimension codes. Adv. Math. Commun. 7(2), 147–160 (2013).
Trautmann A.L., Manganiello F., Braun M., Rosenthal J.: Cyclic orbit codes. IEEE Trans. Inform. Theory 59(11), 7386–7404 (2013).
Xia S., Fu F.: Johnson type bounds on constant dimension codes. Des. Codes Cryptogr. 50(2), 163–172 (2009).
Acknowledgements
The authors would like to express their deepest gratitude to the editor and the anonymous reviewers for their precious comments and valuable suggestions that have helped improve this paper substantially. This work was supported by NSFC (Grant Nos. 11871025, 12271199).
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Communicated by V. A. Zinoviev.
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Li, Y., Liu, H. Cyclic constant dimension subspace codes via the sum of Sidon spaces. Des. Codes Cryptogr. 91, 1193–1207 (2023). https://doi.org/10.1007/s10623-022-01146-9
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DOI: https://doi.org/10.1007/s10623-022-01146-9