Higher Weights of Grassmann Codes

Higher Weights of Grassmann Codes 1

Sudhir R. Ghorpade 2

Department of Mathematics
Indian Institute of Technology, Bombay,
Powai, Mumbai 400076, India

E-Mail: srg@math.iitb.ernet.in


Gilles Lachaud

Équipe ``Arithmétique et Théorie de l'Information''
Institut de Mathématiques de Luminy
Luminy Case 907, 13288 Marseille, Cedex 9, France

E-Mail: lachaud@iml.univ-mrs.fr


Using a combinatorial approach to studying the hyperplane sections of Grassmannians, we give two new proofs of a result of Nogin concerning the higher weights of Grassmann codes. As a consequence, we obtain a bound on the number of higher dimensional subcodes of the Grassmann code having the minimum Hamming norm. We also discuss a generalization of Grassmann codes.


1 Introduction 1
2 Preliminaries 2
3 Linear Sections of Grassmannians 4
4 Computation of Higher Weights 5
5 A Generalization of the Grassmann Code 7
References 9


Linear codes associated to Schubert varieties in Grassmannians, called Schubert codes, were introduced in the above paper. A conjecture, due to the first author, concerning the minimum distance of the Schubert codes was also stated in this paper. For recent progress on this conjecture and some related work, see:

  1. Hao Chen,   On the minimum distances of Schubert codes,
    IEEE Trans. Inform. Theory, Vol. 46 (2000) 1535-1538.
  2. R. Vincenti, On some classical varieties and codes,
    Rapporto Tecnico 20/2000, Dip. Mat., Univ. Perugia, Italy, 2000.
  3. L. Guerra and R. Vincenti, On the linear codes arising from Schubert varieties,
    Des. Codes Cryptogr., Vol. 33 (2004), 173--180.
  4. S. R. Ghorpade and M. A. Tsfasman, Schubert varieties, linear codes and enumerative combinatorics, Finite Fields Appl., (to appear), arXiv.math.CO/0409394.

1 2000 Mathematics Subject Classification. Primary 11T71, 94B05, 94B27 14M12; Secondary 14M15, 51E20.
2 Partially supported by a `Career Award' grant from AICTE, New Delhi and an IRCC grant from IIT Bombay.

This paper is published in: Coding Theory, Cryptography and Related Areas (Guanajuato, 1998), J. Buchmann, T. Hoeholdt, H. Stichtenoth and H. Tapia-Recillas Eds., Springer-Verlag, Berlin (2000), pp. 122-131.

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