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
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.
Contents
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.