On the asymptotic number of non-equivalent q-ary linear codes

Let Mn,q⊂GL(n,Fq) be the group of monomial matrices, i.e., the group generated by all permutation matrices and diagonal matrices in GL(n,Fq). The group Mn,q acts on the set V(Fqn) of all subspaces of Fqn. The number of orbits of this action, denoted by Nn,q, is the number of non-equivalent linear codes in Fqn. It was conjectured by Lax that Nn,q∼|V(Fqn)|n!(q-1)n-1 as n→∞. We confirm this conjecture in this paper.