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

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