Hyperplane sections of Grassmannians and the number of MDS linear codes

We obtain some effective lower and upper bounds for the number of (n, k)-MD S linear codes over F-q. As a consequence, one obtains an asymptotic formul a for this number. These results also apply for the number of inequivalent representations over F-q of the uniform matroid or, alternatively, the numb er of F-q-rational points of certain open strata of Grassmannians. The tech niques used in the determination of bounds for the number of MDS codes are applied to deduce several geometric properties of certain sections of Grass mannians by coordinate hyperplanes. (C) 2001 Academic Press.