August  2017, 11(3): 533-548. doi: 10.3934/amc.2017042

New criteria for MRD and Gabidulin codes and some Rank-Metric code constructions

1. 

Faculty of Mathematics and Statistics, University of St. Gallen, St. Gallen, Switzerland

2. 

Institute of Mathematics, University of Zurich, Zurich, Switzerland

Received  September 2015 Revised  November 2016 Published  August 2017

Fund Project: The authors were partially supported by SNF grant no. 149716.

It is well-known that maximum rank distance (MRD) codes can be constructed as generalized Gabidulin codes. However, it was unknown until recently whether other constructions of linear MRD codes exist. In this paper, we derive a new criterion for linear MRD codes as well as an algebraic criterion for testing whether a given linear MRD code is a generalized Gabidulin code. We then use the criteria to construct examples of linear MRD codes which are not generalized Gabidulin codes.

Citation: 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
References:
[1]

T. P. Berger, Isometries for rank distance and permutation group of Gabidulin codes, IEEE Trans. Inf. Theory, 49 (2003), 3016-3019.  doi: 10.1109/TIT.2003.819322.  Google Scholar

[2]

A. CossidenteG. Marino and F. Pavese, Non-linear maximum rank distance codes, Des. Codes Crypt., 79 (2016), 597-609.  doi: 10.1007/s10623-015-0108-0.  Google Scholar

[3]

J. de la CruzM. KiermaierA. Wassermann and W. Willems, Algebraic structures of MRD codes, Adv. Math. Commun., 10 (2016), 499-510.  doi: 10.3934/amc.2016021.  Google Scholar

[4]

P. Delsarte, Bilinear forms over a finite field, with applications to coding theory, J. Combin. Theory Ser. A, 25 (1978), 226-241.  doi: 10.1016/0097-3165(78)90015-8.  Google Scholar

[5]

E. M. Gabidulin, Theory of codes with maximum rank distance, Probl. Peredachi Inf., 21 (1985), 3-16.   Google Scholar

[6]

E. M. Gabidulin, A. V. Paramonov and O. V. Tretjakov, Ideals over a non-commutative ring and their application in cryptology, in Proc. 10th Ann. Int. Conf. Theory Appl. Crypt. Techn. EUROCRYPT'91, Springer-Verlag, Berlin, 1991,482–489. doi: 10.1007/3-540-46416-6_41.  Google Scholar

[7]

M. Giorgetti and A. Previtali, Galois invariance, trace codes and subfield subcodes, Finite Fields Appl., 16 (2010), 96-99.  doi: 10.1016/j.ffa.2010.01.002.  Google Scholar

[8]

A. Horlemann-Trautmann, K. Marshall and J. Rosenthal, Extension of Overbeck's attack for Gabidulin based cryptosystems Des. Codes Crypt. published online, 2017. doi: 10.1007/s10623-017-0343-7.  Google Scholar

[9]

A. Kshevetskiy and E. Gabidulin, The new construction of rank codes, in Proc. Int. Symp. Inf. Theory (ISIT), 2005,2105–2108. Google Scholar

[10]

R. Lidl and H. Niederreiter, Introduction to Finite Fields and their Applications Cambridge Univ. Press, Cambridge, 1994. doi: 10.1017/CBO9781139172769.  Google Scholar

[11]

P. Loidreau, Designing a rank metric based McEliece cryptosystem, in Proc. 3rd Int. Conf. Post-Quantum Crypt. PQCrypto'10, Springer-Verlag, Berlin, 2010,142–152. doi: 10.1007/978-3-642-12929-2_11.  Google Scholar

[12]

K. Morrison, Equivalence for 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.  Google Scholar

[13]

J. Sheekey, A new family of linear maximum rank distance codes, Adv. Math. Commun., 10 (2016), 475-488.  doi: 10.3934/amc.2016019.  Google Scholar

[14]

Z. -X. Wan, Geometry of Matrices World Scient. , Singapore, 1996. Google Scholar

show all references

References:
[1]

T. P. Berger, Isometries for rank distance and permutation group of Gabidulin codes, IEEE Trans. Inf. Theory, 49 (2003), 3016-3019.  doi: 10.1109/TIT.2003.819322.  Google Scholar

[2]

A. CossidenteG. Marino and F. Pavese, Non-linear maximum rank distance codes, Des. Codes Crypt., 79 (2016), 597-609.  doi: 10.1007/s10623-015-0108-0.  Google Scholar

[3]

J. de la CruzM. KiermaierA. Wassermann and W. Willems, Algebraic structures of MRD codes, Adv. Math. Commun., 10 (2016), 499-510.  doi: 10.3934/amc.2016021.  Google Scholar

[4]

P. Delsarte, Bilinear forms over a finite field, with applications to coding theory, J. Combin. Theory Ser. A, 25 (1978), 226-241.  doi: 10.1016/0097-3165(78)90015-8.  Google Scholar

[5]

E. M. Gabidulin, Theory of codes with maximum rank distance, Probl. Peredachi Inf., 21 (1985), 3-16.   Google Scholar

[6]

E. M. Gabidulin, A. V. Paramonov and O. V. Tretjakov, Ideals over a non-commutative ring and their application in cryptology, in Proc. 10th Ann. Int. Conf. Theory Appl. Crypt. Techn. EUROCRYPT'91, Springer-Verlag, Berlin, 1991,482–489. doi: 10.1007/3-540-46416-6_41.  Google Scholar

[7]

M. Giorgetti and A. Previtali, Galois invariance, trace codes and subfield subcodes, Finite Fields Appl., 16 (2010), 96-99.  doi: 10.1016/j.ffa.2010.01.002.  Google Scholar

[8]

A. Horlemann-Trautmann, K. Marshall and J. Rosenthal, Extension of Overbeck's attack for Gabidulin based cryptosystems Des. Codes Crypt. published online, 2017. doi: 10.1007/s10623-017-0343-7.  Google Scholar

[9]

A. Kshevetskiy and E. Gabidulin, The new construction of rank codes, in Proc. Int. Symp. Inf. Theory (ISIT), 2005,2105–2108. Google Scholar

[10]

R. Lidl and H. Niederreiter, Introduction to Finite Fields and their Applications Cambridge Univ. Press, Cambridge, 1994. doi: 10.1017/CBO9781139172769.  Google Scholar

[11]

P. Loidreau, Designing a rank metric based McEliece cryptosystem, in Proc. 3rd Int. Conf. Post-Quantum Crypt. PQCrypto'10, Springer-Verlag, Berlin, 2010,142–152. doi: 10.1007/978-3-642-12929-2_11.  Google Scholar

[12]

K. Morrison, Equivalence for 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.  Google Scholar

[13]

J. Sheekey, A new family of linear maximum rank distance codes, Adv. Math. Commun., 10 (2016), 475-488.  doi: 10.3934/amc.2016019.  Google Scholar

[14]

Z. -X. Wan, Geometry of Matrices World Scient. , Singapore, 1996. Google Scholar

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