# American Institute of Mathematical Sciences

August  2016, 10(3): 589-600. doi: 10.3934/amc.2016028

## Explicit constructions of some non-Gabidulin linear maximum rank distance codes

 1 Department of Mathematics & Institute of Applied Mathematics, Middle East Technical University, 06800, Ankara, Turkey 2 Department of Mathematics and Institute of Applied Mathematics, Middle East Technical University, Dumlupnar Bulvar, 06800, Ankara

Received  May 2015 Revised  June 2016 Published  August 2016

We investigate rank metric codes using univariate linearized polynomials and multivariate linearized polynomials together. We examine the construction of maximum rank distance (MRD) codes and the test of equivalence between two codes in the polynomial representation. Using this approach, we present new classes of some non-Gabidulin linear MRD codes.
Citation: Kamil Otal, Ferruh Özbudak. Explicit constructions of some non-Gabidulin linear maximum rank distance codes. Advances in Mathematics of Communications, 2016, 10 (3) : 589-600. doi: 10.3934/amc.2016028
##### References:
 [1] J. Berson, Linearized polynomial maps over finite fields,, J. Algebra, 399 (2014), 389. doi: 10.1016/j.jalgebra.2013.10.013. [2] J. de la Cruz, M. Kiermaier, A. Wassermann and W. Willems, Algebraic structures of MRD Codes,, Adv. Math. Commun., 10 (2016), 499. doi: 10.3934/amc.2016021. [3] P. Delsarte, Bilinear forms over a finite field, with applications to coding theory,, J. Comb. Theory A, 25 (1978), 226. doi: 10.1016/0097-3165(78)90015-8. [4] E. M. Gabidulin, Theory of codes with maximum rank distance,, Probl. Inform. Transm., 21 (1985), 1. [5] A.-L. Horlemann-Trautmann and K. Marshall, New criteria for MRD and Gabidulin codes and some rank-metric code constructions,, preprint, (). [6] A. Kshevetskiy and E. Gabidulin, The new construction of rank codes,, in Proc. Int. Symp. Inf. Theory (ISIT 2005), (2005), 2105. [7] R. Lidl and H. Niederreiter, Finite Fields,, Cambridge Univ. Press, (1997). [8] K. Morrison, Equivalence of rank-metric and matrix codes and automorphism groups of Gabidulin codes,, IEEE Trans. Inf. Theory, 60 (2014), 7035. doi: 10.1109/TIT.2014.2359198. [9] O. Ore, On a special class of polynomials,, Trans. Amer. Math. Soc., 35 (1933), 559. doi: 10.2307/1989849. [10] J. Sheekey, A new family of linear maximum rank distance codes,, preprint, ().

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##### References:
 [1] J. Berson, Linearized polynomial maps over finite fields,, J. Algebra, 399 (2014), 389. doi: 10.1016/j.jalgebra.2013.10.013. [2] J. de la Cruz, M. Kiermaier, A. Wassermann and W. Willems, Algebraic structures of MRD Codes,, Adv. Math. Commun., 10 (2016), 499. doi: 10.3934/amc.2016021. [3] P. Delsarte, Bilinear forms over a finite field, with applications to coding theory,, J. Comb. Theory A, 25 (1978), 226. doi: 10.1016/0097-3165(78)90015-8. [4] E. M. Gabidulin, Theory of codes with maximum rank distance,, Probl. Inform. Transm., 21 (1985), 1. [5] A.-L. Horlemann-Trautmann and K. Marshall, New criteria for MRD and Gabidulin codes and some rank-metric code constructions,, preprint, (). [6] A. Kshevetskiy and E. Gabidulin, The new construction of rank codes,, in Proc. Int. Symp. Inf. Theory (ISIT 2005), (2005), 2105. [7] R. Lidl and H. Niederreiter, Finite Fields,, Cambridge Univ. Press, (1997). [8] K. Morrison, Equivalence of rank-metric and matrix codes and automorphism groups of Gabidulin codes,, IEEE Trans. Inf. Theory, 60 (2014), 7035. doi: 10.1109/TIT.2014.2359198. [9] O. Ore, On a special class of polynomials,, Trans. Amer. Math. Soc., 35 (1933), 559. doi: 10.2307/1989849. [10] J. Sheekey, A new family of linear maximum rank distance codes,, preprint, ().
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