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

Let be the group of monomial matrices, i.e., the group generated by all permutation matrices and diagonal matrices in . The group acts on the set of all subspaces of . The number of orbits of this action, denoted by N_n,q, is the number of non-equivalent linear codes in . It was conjectured by Lax that as n→∞. We confirm this conjecture in this paper.