A note on the weight spectrum of the Schubert code \(C_{\alpha }(2, m)\)

  • Fernando L. Piñero
  • Prasant Singh


We consider the Schubert code \(C_{\alpha }(2, m)\) associated to the \(\mathbb {F}_q\)-rational points of the Schubert variety \(\Omega _{\alpha }(2,m)\) in the Grassmannian \(G_{2,m}\). A correspondence between codewords of \(C_{\alpha }(2, m)\) and skew-symmetric matrices of certain special form is given. Using this correspondence, we give a formula for all possible weights of codewords in \(C_{\alpha }(2, m)\). It is shown that the weight of each codeword is divisible by certain power of q. Further, a formula for the weight spectrum of the Schubert code \(C_{\alpha }(2, m)\) is given.


Grassmann code Schubert code Weight spectrum Weight enumerator polynomial 

Mathematics Subject Classification

94B27 11T71 14M15 



The authors would like to thank the anonymous referees for their suggestions to improve the article. We would also like to thank S.R. Ghorpade and A.R. Patil for their warm hospitality.


  1. 1.
    Beelen P., Piñero F.: The structure of dual Grassmann codes. Des. Codes Cryptogr. 79, 451–470 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Beelen P., Ghorpade S.R., Hasan S.U.: Linear codes associated to determinantal varieties. Discret. Math. 338, 1493–1500 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chen H.: On the minimum distance of Schubert codes. IEEE Trans. Inf. Theory 46, 1535–1538 (2000).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ghorpade S.R., Lachaud G.: Higher weights of Grassmann codes. In: Buchmann J., Hoeholdt T., Stichtenoth H., Tapia-Recillas H. (eds.) Coding Theory, Cryptography and Related Areas (Guanajuato, 1998), pp. 122–131. Springer, Berlin (2000).CrossRefGoogle Scholar
  5. 5.
    Ghorpade S.R., Singh P.: Minimum distance and the minimum weight codewords of Schubert codes. Finite Fields Appl. 49, 1–28 (2018).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ghorpade S.R., Tsfasman M.A.: Schubert varieties, linear codes and enumerative combinatorics. Finite Fields Appl. 11, 684–699 (2005).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Guerra L., Vincenti R.: On the linear codes arising from Schubert varieties. Des. Codes Cryptogr. 33, 173–180 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kaipa K., Pillai H.: Weight spectrum of codes associated with the Grassmannian \(G(3,7)\). IEEE Trans. Inf. Theory 59, 983–993 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kleiman S.L., Laksov D.: Schubert calculus. Am. Math. Mon. 79, 1061–1082 (1972).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Nogin D.Y.: Codes associated to Grassmannians. In: Pellikaan R., Perret M., Vlăduţ S.G. (eds.) Arithmetic, Geometry and Coding Theory (Luminy, 1993), pp. 145–154. Walter de Gruyter, Berlin (1996).Google Scholar
  11. 11.
    Nogin D.Y.: The spectrum of codes associated with the Grassmannian variety \(G(3,6)\). Probl. Inf. Transm. 33, 114–123 (1997).MathSciNetzbMATHGoogle Scholar
  12. 12.
    Ryan C.T.: An application of Grassmann varieties to coding theory. Congr. Numer. 57, 257–271 (1987).MathSciNetGoogle Scholar
  13. 13.
    Ryan C.T.: Projective codes based on Grassmann varieties. Congr. Numer. 57, 273–279 (1987).MathSciNetzbMATHGoogle Scholar
  14. 14.
    SageMath: The Sage Mathematics Software System, Version 7.2.3. The Sage Developers (2017). Accessed 01 April 2017.
  15. 15.
    Tsfasman M., Vlăduţ S., Nogin D.: Algebraic Geometric Codes: Basic Notions. Mathematical Surveys and Monographs, vol. 139. American Mathematical Society, Providence (2007).Google Scholar
  16. 16.
    Xiang X.: On the minimum distance conjecture for Schubert codes. IEEE Trans. Inf. Theory 54, 486–488 (2008).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Puerto Rico at PoncePonceUSA
  2. 2.Department of MathematicsIndian Institute of Technology BombayMumbaiIndia

Personalised recommendations