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Bent and vectorial bent functions, partial difference sets, and strongly regular graphs
Self-duality of generalized twisted Gabidulin codes
1. | Department of Mathematics and Institute of Applied Mathematics, Middle East Technical University, 06800, Ankara, Turkey |
2. | Otto-von-Guericke-Universität, Magdeburg, Germany & Universidad del Norte, Barranquilla, Colombia |
Self-duality of Gabidulin codes was investigated in [
References:
[1] |
L. Carlitz,
A note on the Betti-Mathieu group, Portugaliae Math., 22 (1963), 121-125.
|
[2] |
A. Cossidente, G. Marino and F. Pavese,
Non-linear maximum rank distance codes, Des. Codes Cryptogr., 79 (2016), 597-609.
doi: 10.1007/s10623-015-0108-0. |
[3] |
P. Delsarte,
Bilinear forms over a finite field, with applications to coding theory, J. Comb. Theory A, 25 (1978), 226-241.
doi: 10.1016/0097-3165(78)90015-8. |
[4] |
L. E. Dickson,
The analytic representation of substitutions on a power of a prime number of letters with a discussion of the linear group, Ann. Math., 11 (1896), 65-120.
doi: 10.2307/1967217. |
[5] |
N. Durante and A. Siciliano, Non-linear maximum rank distance codes in the cyclic model for the field reduction of finite geometries, Electron. J. Comb., 24 (2017), Paper 2.33, 18 pp. |
[6] |
E. M. Gabidulin,
The theory with maximal rank metric distance, Probl. Inform. Transm., 21 (1985), 1-12.
|
[7] |
A. Kshevetskiy and E. Gabidulin, The new construction of rank codes, Proceedings of Int. Symp. on Inf. Theory, (ISIT 2005), 2105-2108. |
[8] |
R. Lidl and H. Niederreither, Introduction to Finite Fields and Their Applications, Revised Edition, Cambridge University Press, Cambridge, 1994.
doi: 10.1017/CBO9781139172769. |
[9] |
G. Lunardon, R. Trombetti and Y. Zhou, Generalized twisted Gabidulin codes, arXiv: 1507.07855v2. |
[10] |
G. Nebe and W. Willems,
On self-dual MRD codes, Adv. in Math. of Comm., 10 (2016), 633-642.
doi: 10.3934/amc.2016031. |
[11] |
K. Otal and F. Özbudak,
Explicit constructions of some non-Gabidulin linear MRD codes, Adv. in Math. of Comm., 10 (2016), 589-600.
doi: 10.3934/amc.2016028. |
[12] |
K. Otal and F. Özbudak,
Additive rank metric codes, IEEE Trans. Inf. Theory, 63 (2017), 164-168.
doi: 10.1109/TIT.2016.2622277. |
[13] |
K. Otal and F. Özbudak,
Some new non-additive maximum rank distance codes, Finite Fields Appl., 50 (2018), 293-303.
doi: 10.1016/j.ffa.2017.12.003. |
[14] |
A. Ravagnani,
Rank-metric codes and their duality theory, Des. Codes Cryptogr., 80 (2016), 197-216.
doi: 10.1007/s10623-015-0077-3. |
[15] |
J. Sheekey,
A new family of linear maximum rank distance codes, Adv. in Math. of Comm., 10 (2016), 475-488.
doi: 10.3934/amc.2016019. |
[16] |
Z.-X. Wan,
Geometry of Matrices, In memory of Professor L.K. Hua (1910-1985), World Scientific, Singapore, 1996.
doi: 10.1142/9789812830234. |
[17] |
B. Wu and Z. Liu,
Linearized polynomials over finite fields revisited, Finite Fields Appl., 22 (2013), 79-100.
doi: 10.1016/j.ffa.2013.03.003. |
show all references
References:
[1] |
L. Carlitz,
A note on the Betti-Mathieu group, Portugaliae Math., 22 (1963), 121-125.
|
[2] |
A. Cossidente, G. Marino and F. Pavese,
Non-linear maximum rank distance codes, Des. Codes Cryptogr., 79 (2016), 597-609.
doi: 10.1007/s10623-015-0108-0. |
[3] |
P. Delsarte,
Bilinear forms over a finite field, with applications to coding theory, J. Comb. Theory A, 25 (1978), 226-241.
doi: 10.1016/0097-3165(78)90015-8. |
[4] |
L. E. Dickson,
The analytic representation of substitutions on a power of a prime number of letters with a discussion of the linear group, Ann. Math., 11 (1896), 65-120.
doi: 10.2307/1967217. |
[5] |
N. Durante and A. Siciliano, Non-linear maximum rank distance codes in the cyclic model for the field reduction of finite geometries, Electron. J. Comb., 24 (2017), Paper 2.33, 18 pp. |
[6] |
E. M. Gabidulin,
The theory with maximal rank metric distance, Probl. Inform. Transm., 21 (1985), 1-12.
|
[7] |
A. Kshevetskiy and E. Gabidulin, The new construction of rank codes, Proceedings of Int. Symp. on Inf. Theory, (ISIT 2005), 2105-2108. |
[8] |
R. Lidl and H. Niederreither, Introduction to Finite Fields and Their Applications, Revised Edition, Cambridge University Press, Cambridge, 1994.
doi: 10.1017/CBO9781139172769. |
[9] |
G. Lunardon, R. Trombetti and Y. Zhou, Generalized twisted Gabidulin codes, arXiv: 1507.07855v2. |
[10] |
G. Nebe and W. Willems,
On self-dual MRD codes, Adv. in Math. of Comm., 10 (2016), 633-642.
doi: 10.3934/amc.2016031. |
[11] |
K. Otal and F. Özbudak,
Explicit constructions of some non-Gabidulin linear MRD codes, Adv. in Math. of Comm., 10 (2016), 589-600.
doi: 10.3934/amc.2016028. |
[12] |
K. Otal and F. Özbudak,
Additive rank metric codes, IEEE Trans. Inf. Theory, 63 (2017), 164-168.
doi: 10.1109/TIT.2016.2622277. |
[13] |
K. Otal and F. Özbudak,
Some new non-additive maximum rank distance codes, Finite Fields Appl., 50 (2018), 293-303.
doi: 10.1016/j.ffa.2017.12.003. |
[14] |
A. Ravagnani,
Rank-metric codes and their duality theory, Des. Codes Cryptogr., 80 (2016), 197-216.
doi: 10.1007/s10623-015-0077-3. |
[15] |
J. Sheekey,
A new family of linear maximum rank distance codes, Adv. in Math. of Comm., 10 (2016), 475-488.
doi: 10.3934/amc.2016019. |
[16] |
Z.-X. Wan,
Geometry of Matrices, In memory of Professor L.K. Hua (1910-1985), World Scientific, Singapore, 1996.
doi: 10.1142/9789812830234. |
[17] |
B. Wu and Z. Liu,
Linearized polynomials over finite fields revisited, Finite Fields Appl., 22 (2013), 79-100.
doi: 10.1016/j.ffa.2013.03.003. |
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