Frontiers of Mathematics in China >
Constructions of Sidon spaces and cyclic subspace codes
Published date: 15 Apr 2022
Copyright
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
He ZHANG , Xiwang CAO . Constructions of Sidon spaces and cyclic subspace codes[J]. Frontiers of Mathematics in China, 2022 , 17(2) : 275 -288 . DOI: 10.1007/s11464-022-1011-4
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)
|
/
〈 | 〉 |