Hyperplane Sections of Grassmannians and the Number of MDS Linear Codes

Hyperplane Sections of Grassmannians and the Number of MDS Linear Codes

Sudhir R. Ghorpade1

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


We obtain some effective lower and upper bounds for the number of (n,k)-MDS linear codes over GF(q). As a consequence, one obtains an asymptotic formula for this number. These results also apply for the number of inequivalent representations over GF(q) of the uniform matroid, or, alternately, the number of GF(q)-rational points of certain open strata of Grassmannians. The techniques used in the determination of bounds for the number of MDS codes are applied to deduce several geometric properties of certain sections of Grassmannians by coordinate hyperplanes.


1 Introduction 1
2 Preliminaries 6
3 Hyperplane Sections of Grassmannians 9
4 Close Families of k-subsets 13
5 Bounds for the Number of MDS Codes 17
6 Geometric Applications 23
7 Tables 35
References 38

1 Partially supported by a `Career Award' grant from AICTE, New Delhi and an IRCC grant from IIT Bombay.

This paper is published in: Finite Fields and their Applications, Vol. 7, No. 4 (2001), pp. 468--506.

Download the full paper as:

PDF File Postscript File DVI File.

Back to the List of Publications

Back to the Sudhir Ghorpade's Home Page