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  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, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

È. M. Gabidulin, Theory of codes of maximum rank distance, Problemy Peredachi Informatsii, 21 (1985), 3-16.   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. Grothendieck, Éléments de géométrie algèbrique: Ⅱ. Étude globale élémentaire de quelques classes de morphismes, Inst. Hautes Études Sci. Publ. Math., 8 (1961), 5-222.   Google Scholar

[9]

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

[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.  Google Scholar

[11]

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

[12]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

È. M. Gabidulin, Theory of codes of maximum rank distance, Problemy Peredachi Informatsii, 21 (1985), 3-16.   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. Grothendieck, Éléments de géométrie algèbrique: Ⅱ. Étude globale élémentaire de quelques classes de morphismes, Inst. Hautes Études Sci. Publ. Math., 8 (1961), 5-222.   Google Scholar

[9]

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

[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.  Google Scholar

[11]

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

[12]

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

[1]

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

[2]

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

[3]

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

[4]

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

[5]

Tomasz Downarowicz, Yonatan Gutman, Dawid Huczek. Rank as a function of measure. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2741-2750. doi: 10.3934/dcds.2014.34.2741

[6]

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

[7]

Mariantonia Cotronei, Tomas Sauer. Full rank filters and polynomial reproduction. Communications on Pure & Applied Analysis, 2007, 6 (3) : 667-687. doi: 10.3934/cpaa.2007.6.667

[8]

Keisuke Minami, Takahiro Matsuda, Tetsuya Takine, Taku Noguchi. Asynchronous multiple source network coding for wireless broadcasting. Numerical Algebra, Control & Optimization, 2011, 1 (4) : 577-592. doi: 10.3934/naco.2011.1.577

[9]

Min Ye, Alexander Barg. Polar codes for distributed hierarchical source coding. Advances in Mathematics of Communications, 2015, 9 (1) : 87-103. doi: 10.3934/amc.2015.9.87

[10]

David Mieczkowski. The first cohomology of parabolic actions for some higher-rank abelian groups and representation theory. Journal of Modern Dynamics, 2007, 1 (1) : 61-92. doi: 10.3934/jmd.2007.1.61

[11]

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

[12]

Zhouchen Lin. A review on low-rank models in data analysis. Big Data & Information Analytics, 2016, 1 (2&3) : 139-161. doi: 10.3934/bdia.2016001

[13]

Frank Blume. Minimal rates of entropy convergence for rank one systems. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 773-796. doi: 10.3934/dcds.2000.6.773

[14]

Michael Blank. Finite rank approximations of expanding maps with neutral singularities. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 749-762. doi: 10.3934/dcds.2008.21.749

[15]

Keith Burns, Katrin Gelfert. Lyapunov spectrum for geodesic flows of rank 1 surfaces. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1841-1872. doi: 10.3934/dcds.2014.34.1841

[16]

Gábor Székelyhidi, Ben Weinkove. On a constant rank theorem for nonlinear elliptic PDEs. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6523-6532. doi: 10.3934/dcds.2016081

[17]

Ke Wei, Jian-Feng Cai, Tony F. Chan, Shingyu Leung. Guarantees of riemannian optimization for low rank matrix completion. Inverse Problems & Imaging, 2020, 14 (2) : 233-265. doi: 10.3934/ipi.2020011

[18]

Yitong Guo, Bingo Wing-Kuen Ling. Principal component analysis with drop rank covariance matrix. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020072

[19]

Stefan Martignoli, Ruedi Stoop. Phase-locking and Arnold coding in prototypical network topologies. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 145-162. doi: 10.3934/dcdsb.2008.9.145

[20]

Giuseppe Bianchi, Lorenzo Bracciale, Keren Censor-Hillel, Andrea Lincoln, Muriel Médard. The one-out-of-k retrieval problem and linear network coding. Advances in Mathematics of Communications, 2016, 10 (1) : 95-112. doi: 10.3934/amc.2016.10.95

2019 Impact Factor: 0.734

Metrics

  • PDF downloads (29)
  • HTML views (75)
  • Cited by (0)

Other articles
by authors

[Back to Top]