Constructions of Sidon spaces and cyclic subspace codes

Abstract

In this paper, we firstly construct several new kinds of Sidon spaces and Sidon sets by investigating some known results. Secondly, using these Sidon spaces, we will present a construction of cyclic subspace codes with cardinality τ · $\frac{{{q^n} - 1}}{{q - 1}}$ and minimum distance 2k−2, where τ is a positive integer. We furthermore give some cyclic subspace codes with size 2τ · $\frac{{{q^n} - 1}}{{q - 1}}$ and without changing the minimum distance 2k−2.

Key words： Sidon spaces; Sidon sets; cyclic subspace codes; minimum distance

References

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

2 Bachoc C , Serra O , Zémor . An analogue of Vosper’s theorem for extension fields. Math. Proc. Cambridge Philos. Soc., 2017, 163 (3): 423- 452

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

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

5 Chen B , Liu H . Constructions of cyclic constant dimension codes. Des. Codes Cryptogr., 2017, 1- 13

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), 2012, 1463- 1547

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

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

9 Feng T , Wang Y . New constructions of large cyclic subspace codes and Sidon spaces. Discrete Math., 2021, 344 (4): 112273

10 Gluesing-Luerssen H , Lehmann H . Distance Distributions of Cyclic Orbit Codes. Des. Codes Cryptogr., 2021, 89: 447- 470

11 Gluesing-Luerssen H , Morrison K , Troha C . Cyclic orbit codes and stabilizer subfields. Adv. Math. Commun., 2015, 9 (2): 177- 197

12 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, 31–42 Springer, Berlin, 2008

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

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

15 Ötal K , Ozbudak F . Cyclic subspace codes via subspace polynomials. Des. Codes Cryptogr., 2017, 85 (2): 191- 204

16 Raviv N , Tamo I . Cyclic subspace codes and Sidon spaces. IEEE International Symposium on Information Theory, 2017, 784- 788

17 Roth R M , Raviv N , Tamo I . Construction of Sidon spaces with Applications to Coding. IEEE Trans. Inf. Theory, 2018, 64 (6): 4412- 4422

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

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

20 Niu Y , Yue Y . Several kinds of large cyclic subspace codes via Sidon spaces. Discrete Math, 2020, 343 (5): 111788

21 Li Y , Liu H . Cyclic subspace codes via the sum of Sidon spaces. arXiv: 2105.12520 [cs.IT].

22 Zhang H , Cao X . Further constructions of cyclic subspace codes. Cryptogr Commun, 2021, 13: 245- 262

23 Zhao W , Tang X . A characterization of cyclic subspace codes via subspace polynomials. Finite Fields Appl, 2019, 57: 1- 12 (in Chinese)