Schubert Varieties, Linear Codes and Enumerative Combinatorics
Schubert Varieties, Linear Codes and Enumerative Combinatorics
Sudhir R. Ghorpade
1
Department of Mathematics
Indian Institute of Technology Bombay,
Powai, Mumbai 400076, India
E-Mail: srg@math.iitb.ac.in
and
Michael A. Tsfasman
2
Institut de Mathématiques de Luminy,
Case 907, 13288 Marseille, France
and
Independent University of Moscow
and
Dorbushin Math. Lab., Institute for Information Transmission Problems, Moscow
E-Mail: tsfasman@iml.univ-mrs.fr
Abstract
We consider linear error correcting codes associated to higher dimensional
projective varieties defined over a finite field.
The problem of determining the basic parameters of such codes often leads to
some interesting and difficult questions in combinatorics and algebraic
geometry. This is illustrated by codes associated to Schubert varieties in
Grassmannians, called Schubert codes, which have recently been studied.
The basic parameters such as the length, dimension and minimum distance of
these codes are known only in special cases. An upper bound for the minimum
distance is known and it is conjectured that this bound is achieved. We give
explicit formulae for the length and dimension of arbitrary Schubert codes
and prove the minimum distance conjecture in the affirmative for codes
associated to Schubert divisors.
Contents
1 | Introduction | 1 |
2 | Length of Schubert Codes | 4 |
3 | Dimension of Schubert Codes | 7 |
4 | Minimum Distance Conjecture for Schubert Divisors | 9 |
| References | 11 |
1
Partially supported by the IRCC grant 97IR012 from IIT Bombay.
2
Partially supported by the RFBR grants 99-01-01204, 02-01-01041 and 02-01-22005.
This paper is published in
Finite Fields and their
Applications, Vol. 11, No. 4 (2005), pp. 684-699.
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Note:
An extended abstract of this paper appears in the article
Classical varieties, codes and
combinatorics
which may be referred to for a leisurely introduction to this paper.
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