Boletín de la Sociedad Matemática Mexicana

, Volume 25, Issue 3, pp 747–758 | Cite as

Higher weights for the Lagrangian–Grassmannian codes

  • Jesús Carrillo-Pacheco
  • Felipe ZaldívarEmail author
Original Article


We obtain descriptions of the Lagrangian–Grassmannian code (Carrillo-Pacheco and Zaldívar in Des Codes Cryptogr 60:291–268, 2011) as a linear code associated with an FFN(1, q)-projective variety (Carrillo-Pacheco and Zaldívar in Adv Math Commun 10:209–220, 2016), and using these descriptions, we obtain bounds for its higher weights.


Algebraic geometry codes Grassmann codes Lagrangian–Grassmannian codes Higher weights 

Mathematics Subject Classification

Primary: 94B27 Secondary: 14G50 14M15 11T71 



We would like to thank the referees for their valuable comments and suggestions.


  1. 1.
    Ballico, E., Cossidente, A.: On the finite field nullstellensatz. Aust. J. Comb. 21, 57–60 (2000)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Ballico, E., Cossidente, A.: Finite field nullstellensatz and grassmannians. Aust. J. Comb. 24, 313–315 (2001)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Carrillo-Pacheco, J., Zaldívar, F.: On Lagrangian-Grassmannian codes. Designs Codes Cryptogr. 60, 291–298 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Carrillo-Pacheco, J., Vega, G., Zaldívar, F.: The weight distribution of a family of Lagrangian–Grassmannian codes. In: Hajji, El (ed.) C2SI 2015, Lect. Notes Comp. Sci., vol. 9084, pp. 240–246. Springer, Berlin (2015)Google Scholar
  5. 5.
    Carrillo-Pacheco, J., Zaldívar, F.: On Codes over FFN\((1, q)\)-projective varieties. Adv. Math. Commun. 10, 209–220 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Carrillo-Pacheco, J., Jarquín-Zárate, F., Velasco-Fuentes, M., Zaldívar, F.: An explicit description in terms of Plücker Coordinates of the Lagrangian–Grassmannian. Preprint. arXiv:1601.07501
  7. 7.
    Carrillo-Pacheco, J., Jarquín-Zarate, F.: A family of low density matrices in the Lagrangian–Grassmannian. Spec Matrices 2018(6), 237–248 (2018)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Gallager, R.G.: Low density parity check codes. IRE Trans. Inf. Theory 8(1), 21–28 (1962)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Ghorpade, S.R., Lachaud, G.: Higher weights of Grassmann codes. Coding Theory, Cryptography and Related Areas, pp. 122–131. Springer, Berlin (2000)CrossRefGoogle Scholar
  10. 10.
    Ghorpade, S.R., Lachaud, G.: Hyperplane sections of Grassmannians and the number of MDS linear codes. Finite Fields Appl 7, 468–506 (2001)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kreuzer, M., Robbiano, L.: Computational Commutative Algebra 1. Springer, Berlin (2000)CrossRefGoogle Scholar
  12. 12.
    Yu Nogin, D.: Codes associated to Grassmannians. Arithmetic, Geometry and Coding Theory (Luminy 1993), pp. 145–154. Walter de Gruyter, Berlin (1996)Google Scholar
  13. 13.
    Yu Nogin, D.: Spectrum of codes associated with the Grassmannian \(G(3,6)\). Probl. Inf. Transm. 33(2), 114–123 (1997)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Ryan, C.T.: An application of Grassmannian varieties to coding theory. Congr. Numer. 57, 257–271 (1987)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Ryan, C.T.: Projective codes based on Grassmann varieties. Congr. Numer. 57, 273–279 (1987)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Ryan, C.T., Ryan, K.M.: The minimum weight of Grassmannian codes \(C(k, n)\). Discret. Appl. Math. 28, 149–156 (1990)CrossRefGoogle Scholar
  17. 17.
    Tsfasman, M.A., Vladut, S.G.: Geometric approach to higher weights. IEEE Trans. Inf. Theory 41, 1564–1588 (1995)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Tsfasman, M.A., Vladut, S.G., Nogin, D.: Algebraic Geometric Codes: Basic Notions. American Mathematical Society, Providence (2007)CrossRefGoogle Scholar
  19. 19.
    Wei, V.K.: Generalized Hamming weights for linear codes. IEEE Trans. Inf. Theory 37, 1412–1418 (1991)MathSciNetCrossRefGoogle Scholar

Copyright information

© Sociedad Matemática Mexicana 2018

Authors and Affiliations

  1. 1.Academia de MatemáticasUniversidad Autónoma de la Ciudad de MéxicoMexicoMexico
  2. 2.Departamento de MatemáticasUniversidad Autónoma Metropolitana-IMexicoMexico

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