American Institute of Mathematical Sciences

Advanced
  • Previous Article
    The singularity attack to the multivariate signature scheme HIMQ-3
  • AMC Home
  • This Issue
  • Next Article
    Finding small solutions of the equation $ \mathit{{Bx-Ay = z}} $ and its applications to cryptanalysis of the RSA cryptosystem
doi: 10.3934/amc.2020105

Orbit codes from forms on vector spaces over a finite field

Angela Aguglia 1, , Antonio Cossidente 2, , Giuseppe Marino 3, , Francesco Pavese 1,, and  Alessandro Siciliano 2,

1. 

Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, Bari, I-70126, Italy
2. 

Dipartimento di Matematica, Informatica ed Economia, Università degli Studi della Basilicata, Potenza, I-85100, Italy
3. 

Dipartimento di Matematica e Applicazioni "Renato Caccioppoli", Università degli Studi di Napoli "Federico II", Napoli, I-80138, Italy

* Corresponding author: Francesco Pavese

Received  April 2020 Revised  July 2020 Published  August 2020

Fund Project: The research was supported by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA - INdAM)

In this paper we construct different families of orbit codes in the vector spaces of the symmetric bilinear forms, quadratic forms and Hermitian forms on an $ n $-dimensional vector space over the finite field $ {\mathbb F_{q}} $. All these codes admit the general linear group $ {{{{\rm{GL}}}}}(n,q) $ as a transitive automorphism group.

Keywords: Orbit code, Constant dimension code, Subspace code.
Mathematics Subject Classification: Primary: 05B25; Secondary: 15A63, 94B25.
Citation: Angela Aguglia, Antonio Cossidente, Giuseppe Marino, Francesco Pavese, Alessandro Siciliano. Orbit codes from forms on vector spaces over a finite field. Advances in Mathematics of Communications, doi: 10.3934/amc.2020105
References:
[1]

R. Ahlswede, N. Cai, S.-Y. Li and R. W. Yeung, Network information flow, IEEE Trans. Inform. Theory, 46 (2000), 1204-1216.  doi: 10.1109/18.850663.  Google Scholar

[2] M. Aschbacher, Finite Group Theory, Cambridge Studies in Advanced Mathematics, 10. Cambridge University Press, Cambridge, 1986.   Google Scholar
[3]

E. Ben-Sasson, T. Etzion, A. Gabizon and N. Raviv, Subspace polynomials and cyclic subspace codes, IEEE Trans. Inform. Theory, 62 (2016), 1157-1165.  doi: 10.1109/TIT.2016.2520479.  Google Scholar

[4]

O. Bottema, On the Betti-Mathieu group, Nieuw Arch. Wisk., 16 (1930), 46-50.   Google Scholar

[5]

M. Braun, T. Etzion, P. R. J. Östergård, A. Vardy and A. Wassermann, Existence of $q$–analogs of Steiner systems, Forum Math. Pi, 4 (2016), e7, 14 pp. doi: 10.1017/fmp.2016.5.  Google Scholar

[6]

L. Carlitz, A Note on the Betti-Mathieu group, Portugaliae mathematica, 22 (1963), 121-125.   Google Scholar

[7]

B. Chen and H. Liu, Constructions of cyclic constant dimension codes, Des. Codes Cryptogr., 86 (2018), 1267-1279.  doi: 10.1007/s10623-017-0394-9.  Google Scholar

[8]

J.-J. Climent, V. Requena and X. Soler-Escrivà, A construction of Abelian non-cyclic orbit codes, Cryptography and Communication, 11 (2019), 839-852.  doi: 10.1007/s12095-018-0306-5.  Google Scholar

[9]

B. N. Cooperstein, External flats to varieties in ${{{\rm PG}}}(M_{n, n}({{{\rm GF}}}(q)))$, Linear Algebra Appl., 267 (1997), 175-186.   Google Scholar

[10]

A. Cossidente and F. Pavese, On subspace codes, Des. Codes Cryptogr., 78 (2016), 527-531.  doi: 10.1007/s10623-014-0018-6.  Google Scholar

[11]

A. Cossidente and F. Pavese, Veronese subspace codes, Des. Codes Cryptogr., 81 (2016), 445-457.  doi: 10.1007/s10623-015-0166-3.  Google Scholar

[12]

A. Cossidente and F. Pavese, Subspace codes in ${{{\rm PG}}}(2n-1, q)$, Combinatorica, 37 (2017), 1073-1095.  doi: 10.1007/s00493-016-3354-5.  Google Scholar

[13]

A. Cossidente, F. Pavese and L. Storme, Geometrical aspects of subspace codes, in Network Coding and Subspace Designs, 107–129, Signals Commun. Technol., Springer, Cham, 2018.  Google Scholar

[14]

A. Cossidente, F. Pavese and L. Storme, Optimal subspace codes in ${{{\rm PG}}}(4, q)$, Adv. Math. Commun., 13 (2019), 393-404.  doi: 10.3934/amc.2019025.  Google Scholar

[15]

A. Cossidente, S. Kurz, G. Marino and F. Pavese, Combining subspace codes, preprint, arXiv: 1911.03387. Google Scholar

[16]

B. Csajbók and A. Siciliano, Puncturing maximum rank distance codes, J. Algebraic Combin., 49 (2019), 507-534.  doi: 10.1007/s10801-018-0833-3.  Google Scholar

[17]

P. Dembowski, Finite Geometries, Springer-Verlag, Berlin-New York, 1968.  Google Scholar

[18]

N. Durante and A. Siciliano, Non-linear maximum rank distance codes in the cyclic model for the field reduction of finite geometries, Electron. J. Combin., 24 (2017), 18 pp.  Google Scholar

[19]

T. Etzion and N. Silberstein, Error-correcting codes in projective spaces via rank- metric codes and Ferrers diagrams, IEEE Trans. Inform. Theory, 55 (2009), 2909-2919.  doi: 10.1109/TIT.2009.2021376.  Google Scholar

[20]

T. Etzion and N. Silberstein, Codes and designs related to lifted MRD codes, IEEE Trans. Inform. Theory, 59 (2013), 1004-1017.  doi: 10.1109/TIT.2012.2220119.  Google Scholar

[21]

T. Etzion and A. Vardy, Error-correcting codes in projective space, IEEE Trans. Inform. Theory, 57 (2011), 1165-1173.  doi: 10.1109/TIT.2010.2095232.  Google Scholar

[22]

G. Faina, G. Kiss, S. Marcugini and F. Pambianco, The cyclic model for ${{{\rm PG}}}(n-1, q)$ and a construction of arcs, European J. Combin., 23 (2002), 31-35.  doi: 10.1006/eujc.2001.0525.  Google Scholar

[23]

H. Gluesing-Luerssen, K. Morrison and C. Troha, Cyclic orbit codes and stabilizer subfields, Adv. Math. Commun., 9 (2015), 177-197.  doi: 10.3934/amc.2015.9.177.  Google Scholar

[24]

H. Gluesing-Luerssen and C. Troha, Construction of subspace codes through linkage, Adv. Math. Commun., 10 (2016), 525-540.  doi: 10.3934/amc.2016023.  Google Scholar

[25]

D. Heinlein, M. Kiermaier, S. Kurz and A. Wassermann, Tables of subspace codes, preprint, arXiv: 1601.02864, 2016. Google Scholar

[26] J. W. P. Hirschfeld, Projective Geometries Over Finite Fields, 2nd ed, Clarendon Press, Oxford, 1998.   Google Scholar
[27]

T. Ho, M. Médard, R. Koetter, D. R. Karger, M. Effros, J. Shi and B. Leong, A random linear network coding approach to multicast, IEEE Trans. Inform. Theory, 52 (2006), 4413-4430.  doi: 10.1109/TIT.2006.881746.  Google Scholar

[28]

T. Ho, R. Koetter, M. Médard, D. R. Karger and M. Effros, The benefits of coding over routing in a randomized setting, in Proceedings of the 2003 IEEE international symposium on information theory (ISIT 2003), Yokohama, Japan. IEEE, (2003), p442. doi: 10.1109/ISIT.2003.1228459.  Google Scholar

[29]

T. Honold, M. Kiermaier and S. Kurz, Partial spreads and vector space partitions, in Network Coding and Subspace Designs, 131–170, Signals Commun. Technol., Springer, Cham, 2018.  Google Scholar

[30]

A.-L. Horlemann-Trautmann, Message encoding and retrieval for spread and cyclic orbit codes, Des. Codes Cryptogr., 86 (2018), 365-386.  doi: 10.1007/s10623-017-0377-x.  Google Scholar

[31]

A. L. Horlemann-Trautmann and J. Rosenthal, Constructions of constant dimension codes, in Network Coding and Subspace Designs, 25–42, Signals Commun. Technol., Springer, Cham, 2018.  Google Scholar

[32]

B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin-New York, 1967.  Google Scholar

[33]

W. M. Kantor, Linear groups containing a Singer cycle, J. Algebra, 62 (1980), 232-234.  doi: 10.1016/0021-8693(80)90214-8.  Google Scholar

[34]

B. C. Kestenband, Finite projective geometries that are incidence structures of caps, Linear Algebra Appl., 48 (1982), 303-313.  doi: 10.1016/0024-3795(82)90116-1.  Google Scholar

[35]

A. Kohnert and S. Kurz, Construction of large constant-dimension codes with a prescribed minimum distance, Lecture Notes in Computer Science, 5393 (2008), 31-42.  doi: 10.1007/978-3-540-89994-5_4.  Google Scholar

[36]

R. Kötter and F. R. Kschischang, Coding for errors and erasures in random network coding, IEEE Trans. Inform. Theory, 54 (2008), 3579-3591.  doi: 10.1109/TIT.2008.926449.  Google Scholar

[37] R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and its Applications, 20. Cambridge University Press, Cambridge, 1997.   Google Scholar
[38]

K. Otal and F. Özbudak, Cyclic subspace codes via subspace polynomials, Des. Codes Cryptogr., 85 (2017), 191-204.  doi: 10.1007/s10623-016-0297-1.  Google Scholar

[39]

K. Otal and F. Özbudak, Constructions of cyclic subspace codes and maximum rank distance codes, in Network Coding and Subspace Designs, 43–66, Signals Commun. Technol., Springer, Cham, 2018.  Google Scholar

[40]

M. H. Poroch and A. A. Talebi, Product of symplectic groups and its cyclic orbit code, Discrete Math. Algorithms Appl., 11 (2019), 1950061, 25 pp. doi: 10.1142/s1793830919500617.  Google Scholar

[41]

N. Silberstein and A.-L. Trautmann, Subspace codes based on graph matchings, Ferrers diagrams, and pending blocks, IEEE Trans. Inform. Theory, 61 (2015), 3937-3953.  doi: 10.1109/TIT.2015.2435743.  Google Scholar

[42]

D. Silva, F. R. Kschischang and R. Koetter, A rank-metric approach to error control in random network coding, IEEE Trans. Inform. Theory, 54 (2008), 3951-3967.  doi: 10.1109/TIT.2008.928291.  Google Scholar

[43]

J. Singer, A theorem in finite projective geometry and some applications to number theory, Trans. Amer. Math. Soc., 43 (1938), 377-385.  doi: 10.1090/S0002-9947-1938-1501951-4.  Google Scholar

[44]

A.-L. Trautmann, Isometry and automorphisms of constant dimension codes, Adv. Math. Commun., 7 (2013), 147-160.  doi: 10.3934/amc.2013.7.147.  Google Scholar

[45]

A.-L. Trautmann, F. Manganiello, M. Braun and J. Rosenthal, Cyclic orbit codes, IEEE Trans. Inf. Theory, 59 (2013), 7386-7404.  doi: 10.1109/TIT.2013.2274266.  Google Scholar

[46]

A.-L. Trautmann, F. Manganiello and J. Rosenthal, Orbit codes - a new concept in the area of network coding, in Proc. IEEE Inf. Theory Workshop, Dublin, Ireland, 2010, 1–4. doi: 10.1109/CIG.2010.5592788.  Google Scholar

[47]

D. E. Taylor, The Geometry of the Classical Groups, Sigma Series in Pure Mathematics, 9. Heldermann Verlag, Berlin, 1992.  Google Scholar

[48]

Z.-X. Wan, Geometry of matrices, World Scientific Publishing Co. NJ, 1996. doi: 10.1142/9789812830234.  Google Scholar

[49]

B. Wu and Z. Liu, Linearized polynomials over finite fields revisited, Finite Fields Appl., 22 (2013), 79-100.  doi: 10.1016/j.ffa.2013.03.003.  Google Scholar

[50]

S.-T. Xia and F.-W. Fu, Johnson type bounds on constant dimension codes, Des. Codes Cryptogr., 50 (2009), 163-172.  doi: 10.1007/s10623-008-9221-7.  Google Scholar

show all references

References:
[1]

R. Ahlswede, N. Cai, S.-Y. Li and R. W. Yeung, Network information flow, IEEE Trans. Inform. Theory, 46 (2000), 1204-1216.  doi: 10.1109/18.850663.  Google Scholar

[2] M. Aschbacher, Finite Group Theory, Cambridge Studies in Advanced Mathematics, 10. Cambridge University Press, Cambridge, 1986.   Google Scholar
[3]

E. Ben-Sasson, T. Etzion, A. Gabizon and N. Raviv, Subspace polynomials and cyclic subspace codes, IEEE Trans. Inform. Theory, 62 (2016), 1157-1165.  doi: 10.1109/TIT.2016.2520479.  Google Scholar

[4]

O. Bottema, On the Betti-Mathieu group, Nieuw Arch. Wisk., 16 (1930), 46-50.   Google Scholar

[5]

M. Braun, T. Etzion, P. R. J. Östergård, A. Vardy and A. Wassermann, Existence of $q$–analogs of Steiner systems, Forum Math. Pi, 4 (2016), e7, 14 pp. doi: 10.1017/fmp.2016.5.  Google Scholar

[6]

L. Carlitz, A Note on the Betti-Mathieu group, Portugaliae mathematica, 22 (1963), 121-125.   Google Scholar

[7]

B. Chen and H. Liu, Constructions of cyclic constant dimension codes, Des. Codes Cryptogr., 86 (2018), 1267-1279.  doi: 10.1007/s10623-017-0394-9.  Google Scholar

[8]

J.-J. Climent, V. Requena and X. Soler-Escrivà, A construction of Abelian non-cyclic orbit codes, Cryptography and Communication, 11 (2019), 839-852.  doi: 10.1007/s12095-018-0306-5.  Google Scholar

[9]

B. N. Cooperstein, External flats to varieties in ${{{\rm PG}}}(M_{n, n}({{{\rm GF}}}(q)))$, Linear Algebra Appl., 267 (1997), 175-186.   Google Scholar

[10]

A. Cossidente and F. Pavese, On subspace codes, Des. Codes Cryptogr., 78 (2016), 527-531.  doi: 10.1007/s10623-014-0018-6.  Google Scholar

[11]

A. Cossidente and F. Pavese, Veronese subspace codes, Des. Codes Cryptogr., 81 (2016), 445-457.  doi: 10.1007/s10623-015-0166-3.  Google Scholar

[12]

A. Cossidente and F. Pavese, Subspace codes in ${{{\rm PG}}}(2n-1, q)$, Combinatorica, 37 (2017), 1073-1095.  doi: 10.1007/s00493-016-3354-5.  Google Scholar

[13]

A. Cossidente, F. Pavese and L. Storme, Geometrical aspects of subspace codes, in Network Coding and Subspace Designs, 107–129, Signals Commun. Technol., Springer, Cham, 2018.  Google Scholar

[14]

A. Cossidente, F. Pavese and L. Storme, Optimal subspace codes in ${{{\rm PG}}}(4, q)$, Adv. Math. Commun., 13 (2019), 393-404.  doi: 10.3934/amc.2019025.  Google Scholar

[15]

A. Cossidente, S. Kurz, G. Marino and F. Pavese, Combining subspace codes, preprint, arXiv: 1911.03387. Google Scholar

[16]

B. Csajbók and A. Siciliano, Puncturing maximum rank distance codes, J. Algebraic Combin., 49 (2019), 507-534.  doi: 10.1007/s10801-018-0833-3.  Google Scholar

[17]

P. Dembowski, Finite Geometries, Springer-Verlag, Berlin-New York, 1968.  Google Scholar

[18]

N. Durante and A. Siciliano, Non-linear maximum rank distance codes in the cyclic model for the field reduction of finite geometries, Electron. J. Combin., 24 (2017), 18 pp.  Google Scholar

[19]

T. Etzion and N. Silberstein, Error-correcting codes in projective spaces via rank- metric codes and Ferrers diagrams, IEEE Trans. Inform. Theory, 55 (2009), 2909-2919.  doi: 10.1109/TIT.2009.2021376.  Google Scholar

[20]

T. Etzion and N. Silberstein, Codes and designs related to lifted MRD codes, IEEE Trans. Inform. Theory, 59 (2013), 1004-1017.  doi: 10.1109/TIT.2012.2220119.  Google Scholar

[21]

T. Etzion and A. Vardy, Error-correcting codes in projective space, IEEE Trans. Inform. Theory, 57 (2011), 1165-1173.  doi: 10.1109/TIT.2010.2095232.  Google Scholar

[22]

G. Faina, G. Kiss, S. Marcugini and F. Pambianco, The cyclic model for ${{{\rm PG}}}(n-1, q)$ and a construction of arcs, European J. Combin., 23 (2002), 31-35.  doi: 10.1006/eujc.2001.0525.  Google Scholar

[23]

H. Gluesing-Luerssen, K. Morrison and C. Troha, Cyclic orbit codes and stabilizer subfields, Adv. Math. Commun., 9 (2015), 177-197.  doi: 10.3934/amc.2015.9.177.  Google Scholar

[24]

H. Gluesing-Luerssen and C. Troha, Construction of subspace codes through linkage, Adv. Math. Commun., 10 (2016), 525-540.  doi: 10.3934/amc.2016023.  Google Scholar

[25]

D. Heinlein, M. Kiermaier, S. Kurz and A. Wassermann, Tables of subspace codes, preprint, arXiv: 1601.02864, 2016. Google Scholar

[26] J. W. P. Hirschfeld, Projective Geometries Over Finite Fields, 2nd ed, Clarendon Press, Oxford, 1998.   Google Scholar
[27]

T. Ho, M. Médard, R. Koetter, D. R. Karger, M. Effros, J. Shi and B. Leong, A random linear network coding approach to multicast, IEEE Trans. Inform. Theory, 52 (2006), 4413-4430.  doi: 10.1109/TIT.2006.881746.  Google Scholar

[28]

T. Ho, R. Koetter, M. Médard, D. R. Karger and M. Effros, The benefits of coding over routing in a randomized setting, in Proceedings of the 2003 IEEE international symposium on information theory (ISIT 2003), Yokohama, Japan. IEEE, (2003), p442. doi: 10.1109/ISIT.2003.1228459.  Google Scholar

[29]

T. Honold, M. Kiermaier and S. Kurz, Partial spreads and vector space partitions, in Network Coding and Subspace Designs, 131–170, Signals Commun. Technol., Springer, Cham, 2018.  Google Scholar

[30]

A.-L. Horlemann-Trautmann, Message encoding and retrieval for spread and cyclic orbit codes, Des. Codes Cryptogr., 86 (2018), 365-386.  doi: 10.1007/s10623-017-0377-x.  Google Scholar

[31]

A. L. Horlemann-Trautmann and J. Rosenthal, Constructions of constant dimension codes, in Network Coding and Subspace Designs, 25–42, Signals Commun. Technol., Springer, Cham, 2018.  Google Scholar

[32]

B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin-New York, 1967.  Google Scholar

[33]

W. M. Kantor, Linear groups containing a Singer cycle, J. Algebra, 62 (1980), 232-234.  doi: 10.1016/0021-8693(80)90214-8.  Google Scholar

[34]

B. C. Kestenband, Finite projective geometries that are incidence structures of caps, Linear Algebra Appl., 48 (1982), 303-313.  doi: 10.1016/0024-3795(82)90116-1.  Google Scholar

[35]

A. Kohnert and S. Kurz, Construction of large constant-dimension codes with a prescribed minimum distance, Lecture Notes in Computer Science, 5393 (2008), 31-42.  doi: 10.1007/978-3-540-89994-5_4.  Google Scholar

[36]

R. Kötter and F. R. Kschischang, Coding for errors and erasures in random network coding, IEEE Trans. Inform. Theory, 54 (2008), 3579-3591.  doi: 10.1109/TIT.2008.926449.  Google Scholar

[37] R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and its Applications, 20. Cambridge University Press, Cambridge, 1997.   Google Scholar
[38]

K. Otal and F. Özbudak, Cyclic subspace codes via subspace polynomials, Des. Codes Cryptogr., 85 (2017), 191-204.  doi: 10.1007/s10623-016-0297-1.  Google Scholar

[39]

K. Otal and F. Özbudak, Constructions of cyclic subspace codes and maximum rank distance codes, in Network Coding and Subspace Designs, 43–66, Signals Commun. Technol., Springer, Cham, 2018.  Google Scholar

[40]

M. H. Poroch and A. A. Talebi, Product of symplectic groups and its cyclic orbit code, Discrete Math. Algorithms Appl., 11 (2019), 1950061, 25 pp. doi: 10.1142/s1793830919500617.  Google Scholar

[41]

N. Silberstein and A.-L. Trautmann, Subspace codes based on graph matchings, Ferrers diagrams, and pending blocks, IEEE Trans. Inform. Theory, 61 (2015), 3937-3953.  doi: 10.1109/TIT.2015.2435743.  Google Scholar

[42]

D. Silva, F. R. Kschischang and R. Koetter, A rank-metric approach to error control in random network coding, IEEE Trans. Inform. Theory, 54 (2008), 3951-3967.  doi: 10.1109/TIT.2008.928291.  Google Scholar

[43]

J. Singer, A theorem in finite projective geometry and some applications to number theory, Trans. Amer. Math. Soc., 43 (1938), 377-385.  doi: 10.1090/S0002-9947-1938-1501951-4.  Google Scholar

[44]

A.-L. Trautmann, Isometry and automorphisms of constant dimension codes, Adv. Math. Commun., 7 (2013), 147-160.  doi: 10.3934/amc.2013.7.147.  Google Scholar

[45]

A.-L. Trautmann, F. Manganiello, M. Braun and J. Rosenthal, Cyclic orbit codes, IEEE Trans. Inf. Theory, 59 (2013), 7386-7404.  doi: 10.1109/TIT.2013.2274266.  Google Scholar

[46]

A.-L. Trautmann, F. Manganiello and J. Rosenthal, Orbit codes - a new concept in the area of network coding, in Proc. IEEE Inf. Theory Workshop, Dublin, Ireland, 2010, 1–4. doi: 10.1109/CIG.2010.5592788.  Google Scholar

[47]

D. E. Taylor, The Geometry of the Classical Groups, Sigma Series in Pure Mathematics, 9. Heldermann Verlag, Berlin, 1992.  Google Scholar

[48]

Z.-X. Wan, Geometry of matrices, World Scientific Publishing Co. NJ, 1996. doi: 10.1142/9789812830234.  Google Scholar

[49]

B. Wu and Z. Liu, Linearized polynomials over finite fields revisited, Finite Fields Appl., 22 (2013), 79-100.  doi: 10.1016/j.ffa.2013.03.003.  Google Scholar

[50]

S.-T. Xia and F.-W. Fu, Johnson type bounds on constant dimension codes, Des. Codes Cryptogr., 50 (2009), 163-172.  doi: 10.1007/s10623-008-9221-7.  Google Scholar

[1]

Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073
[2]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138
[3]

Annegret Glitzky, Matthias Liero, Grigor Nika. Dimension reduction of thermistor models for large-area organic light-emitting diodes. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020460
[4]

Yuanfen Xiao. Mean Li-Yorke chaotic set along polynomial sequence with full Hausdorff dimension for $ \beta $-transformation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 525-536. doi: 10.3934/dcds.2020267

2019 Impact Factor: 0.734

Tools

Article outline

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences