# American Institute of Mathematical Sciences

August  2016, 10(3): 633-642. doi: 10.3934/amc.2016031

## On self-dual MRD codes

 1 Lehrstuhl D für Mathematik, RWTH Aachen University, 52056 Aachen 2 Fakultät für Mathematik, Otto-von-Guericke Universität, Magdeburg, Germany

Received  May 2015 Revised  December 2015 Published  August 2016

We investigate self-dual MRD codes. In particular we prove that a Gabidulin code in $(\mathbb{F}_q)^{n\times n}$ is equivalent to a self-dual code if and only if its dimension is $n^2/2$, $n \equiv 2 \pmod 4$, and $q \equiv 3 \pmod 4$. On the way we determine the full automorphism group of Gabidulin codes in $(\mathbb{F}_q)^{n\times n}$.
Citation: Gabriele Nebe, Wolfgang Willems. On self-dual MRD codes. Advances in Mathematics of Communications, 2016, 10 (3) : 633-642. doi: 10.3934/amc.2016031
##### References:
 [1] T. Berger, Isometries for rank distance and permutation group of Gabidulin codes, in Proc. ACCT'8, St Petersbourg, 2002, 30-33. doi: 10.1109/TIT.2003.819322. [2] 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. [3] E. Gabidulin, Theory of codes with maximum rank distance, Probl. Inf. Transm., 21 (1985), 1-12. [4] B. Huppert, Endliche Gruppen I, Springer-Verlag, 1967. [5] A. Lempel and G. Seroussi, Factorization of symmetric matrices and trace-orthogonal bases in finite fields, SIAM J. Comput., 9 (1980), 758-767. doi: 10.1137/0209059. [6] R. Lidl and H. Niederreiter, Introduction to Finite Fields and their Applications, Cambridge Univ. Press, 1994. doi: 10.1017/CBO9781139172769. [7] K. Morrison, Equivalence for rank-metric and matrix codes and automorphism groups of Gabidulin codes, IEEE Trans. Inform. Theory, 60 (2014), 7035-7046. doi: 10.1109/TIT.2014.2359198. [8] K. Morrison, An enumeration of the equivalence classes of self-dual matrix codes, Adv. Math. Commun., 9 (2015), 415-436. doi: 10.3934/amc.2015.9.415. [9] A. Ravagnani, Rank-metric codes and their duality theory, Des. Codes Cryptogr., 80 (2016), 197-216. doi: 10.1007/s10623-015-0077-3. [10] W. Scharlau, Quadratic and Hermitian Forms, Grundlehren der mathematischen Wissenschaften 270, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-642-69971-9. [11] J. Sheekey, A new family of linear maximum rank distance codes,, preprint, (). [12] Z.-X. Wan, Geometry of Matrices, World Scientific, Singapore, 1996. doi: 10.1142/9789812830234.

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##### References:
 [1] T. Berger, Isometries for rank distance and permutation group of Gabidulin codes, in Proc. ACCT'8, St Petersbourg, 2002, 30-33. doi: 10.1109/TIT.2003.819322. [2] 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. [3] E. Gabidulin, Theory of codes with maximum rank distance, Probl. Inf. Transm., 21 (1985), 1-12. [4] B. Huppert, Endliche Gruppen I, Springer-Verlag, 1967. [5] A. Lempel and G. Seroussi, Factorization of symmetric matrices and trace-orthogonal bases in finite fields, SIAM J. Comput., 9 (1980), 758-767. doi: 10.1137/0209059. [6] R. Lidl and H. Niederreiter, Introduction to Finite Fields and their Applications, Cambridge Univ. Press, 1994. doi: 10.1017/CBO9781139172769. [7] K. Morrison, Equivalence for rank-metric and matrix codes and automorphism groups of Gabidulin codes, IEEE Trans. Inform. Theory, 60 (2014), 7035-7046. doi: 10.1109/TIT.2014.2359198. [8] K. Morrison, An enumeration of the equivalence classes of self-dual matrix codes, Adv. Math. Commun., 9 (2015), 415-436. doi: 10.3934/amc.2015.9.415. [9] A. Ravagnani, Rank-metric codes and their duality theory, Des. Codes Cryptogr., 80 (2016), 197-216. doi: 10.1007/s10623-015-0077-3. [10] W. Scharlau, Quadratic and Hermitian Forms, Grundlehren der mathematischen Wissenschaften 270, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-642-69971-9. [11] J. Sheekey, A new family of linear maximum rank distance codes,, preprint, (). [12] Z.-X. Wan, Geometry of Matrices, World Scientific, Singapore, 1996. doi: 10.1142/9789812830234.
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