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



Michael A. Tsfasman


Institut de Mathématiques de Luminy, Case 907, 13288 Marseille, France
Independent University of Moscow
Dorbushin Math. Lab., Institute for Information Transmission Problems, Moscow


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.


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.

Download the full paper as:

PDF File Postscript File DVI File.


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.

Back to the List of Publications

Back to the Sudhir Ghorpade's Home Page