Linear Sections of the Grassmannian over Finite Fields

We start with an asymptotic formula for the cardinality of an arbitrary hyperplane section of theGrassmann variety over a finite field with $q$ elements. This formula has bearing on three interesting problems:
1) We present a new asymptotic formula in $q$ for the number of [n,k]_q MDS codes.
2) Higher weights of Grassmann codes: For each number j, what is the maximum number of points n(j) that can lie on the intersection of the Grassmannian with a j dimensional subspace of the Plucker space? This is an open problem except for very small or large j. We report on recent progress on this problem.
3) Is every hyperplane section of the Grassmannian over ar F_p a normal variety? We report on progress on this implied by the estimate above, together with a generalization due to Skorobogatov of the Lefschetz hyperplane theorem for singular varieties in ell-adic cohomology.