Hyperplane sections of Grassmannians and the number of MDS linear codes

Abstract

We obtain some effective lower and upper bounds for the number of (n, k)-MDS linear codes over F-q. As a consequence, one obtains an asymptotic formula for this number. These results also apply for the number of inequivalent representations over F-q of the uniform matroid or, alternatively, the number of F-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.