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Finite Fields and Their Applications
S. R. Ghorpade and G. Lachaud
MDS Codes and Grassmannians
Department of Mathematics,
Indian Institute of Technology, Bombay,
Powai, Mumbai 400076, India
et Théorie de l'Information'', Institut de Mathématiques
de Luminy, Luminy Case 907, 13288 Marseille, Cedex 9, France
We obtain some effective lower and upper bounds for the number of
)-MDS linear codes over
As a consequence, one obtains an asymptotic
formula for this number. These results also apply for the number of inequivalent
of the uniform matroid, or, alternately, the number
-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.