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

Previous Article
The non-existence of $(104,22;3,5)$-arcs
AMC Home
This Issue
Next Article
There is no $[24,12,9]$ doubly-even self-dual code over $\mathbb F_4$

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

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

Kamil Otal ^1, and Ferruh Özbudak ^2,

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, Dumlupnar Bulvar, 06800, Ankara

Received May 2015 Revised June 2016 Published August 2016

Abstract
Related Papers

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.

Keywords: Rank metric codes, generalized Gabidulin codes., maximum rank distance (MRD) codes.

Mathematics Subject Classification: Primary: 94B27, 11T71, 11T55; Secondary: 11T0.

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), 389-406. 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), 499-510. 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), 226-241. doi: 10.1016/0097-3165(78)90015-8.
[4]	E. M. Gabidulin, Theory of codes with maximum rank distance, Probl. Inform. Transm., 21 (1985), 1-12.
[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), 2105-2108.
[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), 7035-7046. doi: 10.1109/TIT.2014.2359198.
[9]	O. Ore, On a special class of polynomials, Trans. Amer. Math. Soc., 35 (1933), 559-584. doi: 10.2307/1989849.
[10]	J. Sheekey, A new family of linear maximum rank distance codes,, preprint, ().

show all references

References:

[1]	J. Berson, Linearized polynomial maps over finite fields, J. Algebra, 399 (2014), 389-406. 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), 499-510. 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), 226-241. doi: 10.1016/0097-3165(78)90015-8.
[4]	E. M. Gabidulin, Theory of codes with maximum rank distance, Probl. Inform. Transm., 21 (1985), 1-12.
[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), 2105-2108.
[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), 7035-7046. doi: 10.1109/TIT.2014.2359198.
[9]	O. Ore, On a special class of polynomials, Trans. Amer. Math. Soc., 35 (1933), 559-584. doi: 10.2307/1989849.
[10]	J. Sheekey, A new family of linear maximum rank distance codes,, preprint, ().

[1]	Anna-Lena Horlemann-Trautmann, Kyle Marshall. New criteria for MRD and Gabidulin codes and some Rank-Metric code constructions. Advances in Mathematics of Communications, 2017, 11 (3) : 533-548. doi: 10.3934/amc.2017042
[2]	John Sheekey. A new family of linear maximum rank distance codes. Advances in Mathematics of Communications, 2016, 10 (3) : 475-488. doi: 10.3934/amc.2016019
[3]	Umberto Martínez-Peñas. Rank equivalent and rank degenerate skew cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 267-282. doi: 10.3934/amc.2017018
[4]	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
[5]	Javier de la Cruz, Michael Kiermaier, Alfred Wassermann, Wolfgang Willems. Algebraic structures of MRD codes. Advances in Mathematics of Communications, 2016, 10 (3) : 499-510. doi: 10.3934/amc.2016021
[6]	Rakhi Pratihar, Tovohery Hajatiana Randrianarisoa. Constructions of optimal rank-metric codes from automorphisms of rational function fields. Advances in Mathematics of Communications, 2022 doi: 10.3934/amc.2022034
[7]	Enhui Lim, Frédérique Oggier. On the generalised rank weights of quasi-cyclic codes. Advances in Mathematics of Communications, 2022 doi: 10.3934/amc.2022010
[8]	Gabriele Nebe, Wolfgang Willems. On self-dual MRD codes. Advances in Mathematics of Communications, 2016, 10 (3) : 633-642. doi: 10.3934/amc.2016031
[9]	Ernst M. Gabidulin, Pierre Loidreau. Properties of subspace subcodes of Gabidulin codes. Advances in Mathematics of Communications, 2008, 2 (2) : 147-157. doi: 10.3934/amc.2008.2.147
[10]	Olof Heden, Martin Hessler. On linear equivalence and Phelps codes. Addendum. Advances in Mathematics of Communications, 2011, 5 (3) : 543-546. doi: 10.3934/amc.2011.5.543
[11]	Diego Napp, Roxana Smarandache. Constructing strongly-MDS convolutional codes with maximum distance profile. Advances in Mathematics of Communications, 2016, 10 (2) : 275-290. doi: 10.3934/amc.2016005
[12]	José Joaquín Bernal, Diana H. Bueno-Carreño, Juan Jacobo Simón. Cyclic and BCH codes whose minimum distance equals their maximum BCH bound. Advances in Mathematics of Communications, 2016, 10 (2) : 459-474. doi: 10.3934/amc.2016018
[13]	Olof Heden, Denis S. Krotov. On the structure of non-full-rank perfect $q$-ary codes. Advances in Mathematics of Communications, 2011, 5 (2) : 149-156. doi: 10.3934/amc.2011.5.149
[14]	Tim Alderson, Alessandro Neri. Maximum weight spectrum codes. Advances in Mathematics of Communications, 2019, 13 (1) : 101-119. doi: 10.3934/amc.2019006
[15]	José Ignacio Iglesias Curto. Generalized AG convolutional codes. Advances in Mathematics of Communications, 2009, 3 (4) : 317-328. doi: 10.3934/amc.2009.3.317
[16]	Yujuan Li, Guizhen Zhu. On the error distance of extended Reed-Solomon codes. Advances in Mathematics of Communications, 2016, 10 (2) : 413-427. doi: 10.3934/amc.2016015
[17]	Carlos Munuera, Morgan Barbier. Wet paper codes and the dual distance in steganography. Advances in Mathematics of Communications, 2012, 6 (3) : 273-285. doi: 10.3934/amc.2012.6.273
[18]	San Ling, Buket Özkaya. New bounds on the minimum distance of cyclic codes. Advances in Mathematics of Communications, 2021, 15 (1) : 1-8. doi: 10.3934/amc.2020038
[19]	Jinmei Fan, Yanhai Zhang. Optimal quinary negacyclic codes with minimum distance four. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2021043
[20]	Olof Heden, Fabio Pasticci, Thomas Westerbäck. On the symmetry group of extended perfect binary codes of length $n+1$ and rank $n-\log(n+1)+2$. Advances in Mathematics of Communications, 2012, 6 (2) : 121-130. doi: 10.3934/amc.2012.6.121

2020 Impact Factor: 0.935

Tools

Metrics

Other articles
by authors

on AIMS
Kamil Otal Ferruh Özbudak
on Google Scholar
Kamil Otal Ferruh Özbudak

[Back to Top]