Hyperplane Sections of Grassmannians and the Number of MDS Linear Codes

Authors: Ghorpade S.R.¹; Lachaud G.²

Source: Finite Fields and Their Applications, Volume 7, Number 4, October 2001 , pp. 468-506(39)

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Abstract:

We obtain some effective lower and upper bounds for the number of (n,k)-MDS linear codes over Fopf _q. As a consequence, one obtains an asymptotic formula for this number. These results also apply for the number of inequivalent representations over Fopf _q of the uniform matroid or, alternately, the number of Fopf _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. Copyright 2001 Academic Press.

Keywords: Grassmannians; MDS codes; hyperplane sections; uniform matroids; arcs in projective spaces

Language: English

Document Type: Research article

Affiliations: 1: Department of Mathematics, Indian Institute of Technology, Bombay, Powai, Mumbai, 400076, India 2: Institut de Mathématiques de Luminy, Luminy Case 907, Marseille, Cedex 9, 13288, France

Publication date: 2001-10-01