# American Institute of Mathematical Sciences

November 2018, 12(4): 707-721. doi: 10.3934/amc.2018042

## 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

* Corresponding author: Ferruh Özbudak

The current affilitaion is: TÜBİTAK BİLGEM UEKAE, 41470, Gebze/Kocaeli, Turkey

Received  June 2017 Revised  February 2018 Published  September 2018

Self-duality of Gabidulin codes was investigated in [10] and the authors provided an if and only if condition for a Gabidulin code to be equivalent to a self-dual maximum rank distance (MRD) code. In this paper, we investigate the analog problem for generalized twisted Gabidulin codes (a larger family of linear MRD codes including the family of Gabidulin codes). We observe that the condition presented in [10] still holds for generalized Gabidulin codes (an intermediate family between Gabidulin codes and generalized twisted Gabidulin codes). However, beyond the family of generalized Gabidulin codes we observe that some additional conditions are required depending on the additional parameters. Our tools are similar to those in [10] but we also use linearized polynomials, which leads to further tools and direct proofs.

Citation: Kamil Otal, Ferruh Özbudak, Wolfgang Willems. Self-duality of generalized twisted Gabidulin codes. Advances in Mathematics of Communications, 2018, 12 (4) : 707-721. doi: 10.3934/amc.2018042
##### 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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##### 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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