November  2021, 15(4): 575-587. doi: 10.3934/amc.2020083

Rank weights for arbitrary finite field extensions

Institut Fourier, CS 40700, 38058 Grenoble Cedex 9, France

* Corresponding author: Grégory Berhuy

Received  February 2019 Revised  December 2019 Published  November 2021 Early access  June 2020

In this paper, we study several definitions of generalized rank weights for arbitrary finite extensions of fields. We prove that all these definitions coincide, generalizing known results for extensions of finite fields.

Citation: Grégory Berhuy, Jean Fasel, Odile Garotta. Rank weights for arbitrary finite field extensions. Advances in Mathematics of Communications, 2021, 15 (4) : 575-587. doi: 10.3934/amc.2020083
References:
[1]

D. Augot, P. Loidreau and G. Robert, Rank metric and Gabidulin codes in characteristic zero, in 2013 IEEE International Symposium on Information Theory, (2013), 509–513. doi: 10.1109/ISIT.2013.6620278.

[2]

S. Bosch, W. Lütkebohmert and M. Raynaud, Néron Models. Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 3, Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-642-51438-8.

[3]

B. Conrad, O. Gabber and G. Prasad, Pseudo-Reductive Groups, New Mathematical Monographs, Vol. 26, Cambridge University Press, Cambridge, 2015. doi: 10.1017/CBO9781316092439.

[4]

P. Delsarte, On subfield codes of modified Reed-Solomon codes, IEEE Trans. Inform. Theory IT-21, (1975), no. 5,575–576. doi: 10.1109/tit.1975.1055435.

[5]

J. Ducoat, Generalized rank weights: A duality statement. Topics in Finite Fields, in Contemporary Mathematics, Vol. 632, Amer. Math. Soc., Providence, RI, 2015,101–109. doi: 10.1090/conm/632/12622.

[6]

È. M. Gabidulin, Theory of codes of maximum rank distance, Problemy Peredachi Informatsii, 21 (1985), 3-16. 

[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.

[8]

A. Grothendieck, Éléments de géométrie algèbrique: Ⅱ. Étude globale élémentaire de quelques classes de morphismes, Inst. Hautes Études Sci. Publ. Math., 8 (1961), 5-222. 

[9]

R. Jurrius and G. R. Pellikaan, On defining generalized rank weights, Adv. Math. Commun., 11 (2017), 225-235.  doi: 10.3934/amc.2017014.

[10]

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.

[11]

F. Oggier and A. Sboui, On the existence of generalized rank weights, in 2012 IEEE International Symposium on Information Theory, (2012), 406–410.

[12]

B. Poonen, Rational Points on Varieties, Graduate Studies in Mathematics, Vol. 186, American Mathematical Society, Providence, RI, 2017.

show all references

References:
[1]

D. Augot, P. Loidreau and G. Robert, Rank metric and Gabidulin codes in characteristic zero, in 2013 IEEE International Symposium on Information Theory, (2013), 509–513. doi: 10.1109/ISIT.2013.6620278.

[2]

S. Bosch, W. Lütkebohmert and M. Raynaud, Néron Models. Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 3, Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-642-51438-8.

[3]

B. Conrad, O. Gabber and G. Prasad, Pseudo-Reductive Groups, New Mathematical Monographs, Vol. 26, Cambridge University Press, Cambridge, 2015. doi: 10.1017/CBO9781316092439.

[4]

P. Delsarte, On subfield codes of modified Reed-Solomon codes, IEEE Trans. Inform. Theory IT-21, (1975), no. 5,575–576. doi: 10.1109/tit.1975.1055435.

[5]

J. Ducoat, Generalized rank weights: A duality statement. Topics in Finite Fields, in Contemporary Mathematics, Vol. 632, Amer. Math. Soc., Providence, RI, 2015,101–109. doi: 10.1090/conm/632/12622.

[6]

È. M. Gabidulin, Theory of codes of maximum rank distance, Problemy Peredachi Informatsii, 21 (1985), 3-16. 

[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.

[8]

A. Grothendieck, Éléments de géométrie algèbrique: Ⅱ. Étude globale élémentaire de quelques classes de morphismes, Inst. Hautes Études Sci. Publ. Math., 8 (1961), 5-222. 

[9]

R. Jurrius and G. R. Pellikaan, On defining generalized rank weights, Adv. Math. Commun., 11 (2017), 225-235.  doi: 10.3934/amc.2017014.

[10]

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.

[11]

F. Oggier and A. Sboui, On the existence of generalized rank weights, in 2012 IEEE International Symposium on Information Theory, (2012), 406–410.

[12]

B. Poonen, Rational Points on Varieties, Graduate Studies in Mathematics, Vol. 186, American Mathematical Society, Providence, RI, 2017.

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