February  2017, 11(1): 225-235. doi: 10.3934/amc.2017014

On defining generalized rank weights

1. 

Institut de Mathématiques, Université de Neuchatel, Rue Emilie-Argand 11,2000 Neuchatel, Switzerland

2. 

Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513,5600 MB Eindhoven, The Netherlands

* Corresponding author

Received  August 2015 Published  February 2017

This paper investigates the generalized rank weights, with a definition implied by the study of the generalized rank weight enumerator. We study rank metric codes over $L$, where $L$ is a finite extension of a field $K$. This is a generalization of the case where $K=\mathbb{F}_q $ and $L={\mathbb{F}_{{q^m}}}$ of Gabidulin codes to arbitrary characteristic. We show equivalence to previous definitions, in particular the ones by Kurihara-Matsumoto-Uyematsu [12,13], Oggier-Sboui [16] and Ducoat [6]. As an application of the notion of generalized rank weights, we discuss codes that are degenerate with respect to the rank metric.

Citation: Relinde Jurrius, Ruud Pellikaan. On defining generalized rank weights. Advances in Mathematics of Communications, 2017, 11 (1) : 225-235. doi: 10.3934/amc.2017014
References:
[1]

D. Augot, Generalization of Gabidulin codes over rational function fields, in MTNS-2014 21st Int. Syp. Math. Theory Netw. Syst. , 2014. Google Scholar

[2]

D. Augot, P. Loidreau and G. Robert, Rank metric and Gabidulin codes in characteristic zero, in IEEE ISIT-2013 Int. Syp. Inf. Theory, 2013,509-513. doi: 10.1109/ISIT.2013.6620278.  Google Scholar

[3]

T. Berger, Isometries for rank distance and permutation group of Gabidulin codes, in Proc. 8th Int. Workshop Algebr. Combin. Coding Theory, 2002, 30-33. doi: 10.1109/TIT.2003.819322.  Google Scholar

[4]

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

[5]

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

[6]

J. Ducoat, Generalized rank weights: a duality statement, in Topics in Finite Fields (eds. G. Kyureghyan, G. L. Mullen and A. Pott), AMS, 2015,101-109. doi: 10.1090/conm/632/12622.  Google Scholar

[7]

È. M. Gabidulin, Theory of codes with maximum rank distance, Probl. Pered. Inform., 21 (1985), 3-16.   Google Scholar

[8]

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

[9]

R. Jurrius and R. Pellikaan, Codes, arrangements and matroids, in Algebraic Geometry Modeling in Information Theory (ed. E. Martínez-Moro), World Scientific, New Jersey, 2013,219-325. doi: 10.1142/9789814335768_0006.  Google Scholar

[10]

R. Jurrius and R. Pellikaan, The extended and generalized rank weight enumerator, in Proc. ACA 2014 Appl. Comp. Algebra CACTC@ACA Comp. Algebra Coding Theory Crypt. , Fordham Univ. , New York, 2014. doi: 10.1145/2768577.2768605.  Google Scholar

[11]

G. Katsman and M. Tsfasman, Spectra of algebraic-geometric codes, Probl. Pered. Inform., 23 (1987), 19-34.   Google Scholar

[12]

J. Kurihara, R. Matsumoto and T. Uyematsu, New parameters of linear codes expressing security performance of universal secure network coding, in 50th Ann. Allerton Conf. Commun. Contr. Comp. , 2012,533-540. doi: 10.1109/Allerton.2012.6483264.  Google Scholar

[13]

J. KuriharaR. Matsumoto and T. Uyematsu, Relative generalized rank weight of linear codes and its applications to network coding, IEEE Trans. Inform. Theory, 61 (2015), 3912-3936.  doi: 10.1109/TIT.2015.2429713.  Google Scholar

[14]

S. Lang, Algebra, Addison-Wesley, Reading, 1965.  Google Scholar

[15]

U. Martínez-Peñas, On the Similarities Between Generalized Rank and Hamming Weights and Their Applications to Network Coding, IEEE Trans. Inform. Theory, 62 (2016), 4081-4095.  doi: 10.1109/TIT.2016.2570238.  Google Scholar

[16]

F. Oggier and A. Sboui, On the existence of generalized rank weights, in IEEE ISIT-2012 Int. Symp. Inform. Theory, 2012,406-410. Google Scholar

[17]

A. Ravagnani, Generalized weights: an anticode approach, J. Pure Appl. Algebra, 220 (2016), 1946-1962.  doi: 10.1016/j.jpaa.2015.10.009.  Google Scholar

[18]

H. Stichtenoth, On the dimension of subfield subcodes, IEEE Trans. Inform. Theory, 36 (1990), 90-93.  doi: 10.1109/18.50376.  Google Scholar

show all references

References:
[1]

D. Augot, Generalization of Gabidulin codes over rational function fields, in MTNS-2014 21st Int. Syp. Math. Theory Netw. Syst. , 2014. Google Scholar

[2]

D. Augot, P. Loidreau and G. Robert, Rank metric and Gabidulin codes in characteristic zero, in IEEE ISIT-2013 Int. Syp. Inf. Theory, 2013,509-513. doi: 10.1109/ISIT.2013.6620278.  Google Scholar

[3]

T. Berger, Isometries for rank distance and permutation group of Gabidulin codes, in Proc. 8th Int. Workshop Algebr. Combin. Coding Theory, 2002, 30-33. doi: 10.1109/TIT.2003.819322.  Google Scholar

[4]

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

[5]

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

[6]

J. Ducoat, Generalized rank weights: a duality statement, in Topics in Finite Fields (eds. G. Kyureghyan, G. L. Mullen and A. Pott), AMS, 2015,101-109. doi: 10.1090/conm/632/12622.  Google Scholar

[7]

È. M. Gabidulin, Theory of codes with maximum rank distance, Probl. Pered. Inform., 21 (1985), 3-16.   Google Scholar

[8]

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

[9]

R. Jurrius and R. Pellikaan, Codes, arrangements and matroids, in Algebraic Geometry Modeling in Information Theory (ed. E. Martínez-Moro), World Scientific, New Jersey, 2013,219-325. doi: 10.1142/9789814335768_0006.  Google Scholar

[10]

R. Jurrius and R. Pellikaan, The extended and generalized rank weight enumerator, in Proc. ACA 2014 Appl. Comp. Algebra CACTC@ACA Comp. Algebra Coding Theory Crypt. , Fordham Univ. , New York, 2014. doi: 10.1145/2768577.2768605.  Google Scholar

[11]

G. Katsman and M. Tsfasman, Spectra of algebraic-geometric codes, Probl. Pered. Inform., 23 (1987), 19-34.   Google Scholar

[12]

J. Kurihara, R. Matsumoto and T. Uyematsu, New parameters of linear codes expressing security performance of universal secure network coding, in 50th Ann. Allerton Conf. Commun. Contr. Comp. , 2012,533-540. doi: 10.1109/Allerton.2012.6483264.  Google Scholar

[13]

J. KuriharaR. Matsumoto and T. Uyematsu, Relative generalized rank weight of linear codes and its applications to network coding, IEEE Trans. Inform. Theory, 61 (2015), 3912-3936.  doi: 10.1109/TIT.2015.2429713.  Google Scholar

[14]

S. Lang, Algebra, Addison-Wesley, Reading, 1965.  Google Scholar

[15]

U. Martínez-Peñas, On the Similarities Between Generalized Rank and Hamming Weights and Their Applications to Network Coding, IEEE Trans. Inform. Theory, 62 (2016), 4081-4095.  doi: 10.1109/TIT.2016.2570238.  Google Scholar

[16]

F. Oggier and A. Sboui, On the existence of generalized rank weights, in IEEE ISIT-2012 Int. Symp. Inform. Theory, 2012,406-410. Google Scholar

[17]

A. Ravagnani, Generalized weights: an anticode approach, J. Pure Appl. Algebra, 220 (2016), 1946-1962.  doi: 10.1016/j.jpaa.2015.10.009.  Google Scholar

[18]

H. Stichtenoth, On the dimension of subfield subcodes, IEEE Trans. Inform. Theory, 36 (1990), 90-93.  doi: 10.1109/18.50376.  Google Scholar

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