On the character of Sn acting on subspaces of

Let denote the finite field with q elements. The symmetric group Sn acts on the set of all linear subspaces of by permuting coordinates. We show that the normalized character of the associated permutation representation of Sn approaches the trivial character as n approaches infinity. We also consider the case of the wreath product of and Sn acting on , and formulate a conjecture that would yield the asymptotic number of distinct equivalence classes of q-ary linear codes.