Self-duality of generalized twisted Gabidulin codes

Kamil Otal; Ferruh Özbudak; Wolfgang Willems

doi:10.3934/amc.2018042

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2018, Volume 12, Issue 4: 707-721. Doi: 10.3934/amc.2018042

This issue Previous Article Bent and vectorial bent functions, partial difference sets, and strongly regular graphs Next Article A class of skew-cyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$ with derivation

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
* Corresponding author: Ferruh Özbudak

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

Received: May 31, 2017

Revised: January 31, 2018

Published: October 2018

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Abstract

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.

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Mathematics Subject Classification: Primary: 11T71; Secondary: 94B05.

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