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
and
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
Abstract
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.
Contents
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 |
Afternotes
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:
-
Hao Chen,
On the minimum distances of Schubert codes,
IEEE Trans. Inform. Theory, Vol. 46 (2000) 1535-1538.
-
R. Vincenti, On some classical varieties and
codes,
Rapporto Tecnico 20/2000, Dip. Mat., Univ. Perugia, Italy, 2000.
-
L. Guerra and R. Vincenti, On the linear codes arising from Schubert varieties,
Des. Codes Cryptogr., Vol. 33 (2004), 173--180.
-
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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