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

Let M-n,M-q subset of 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,M-q acts on the set V(F-q(n)) of all subspaces of H. The number of orbits of this action, denoted by N-n,N-q, is the number of non-equivalent linear codes in F-q(n). It was conjectured by Lax that N-n,N-q similar to vertical bar V(F-q(n))vertical bar/n!(q - 1)(n-1) as n -> infinity. We confirm this conjecture in this paper. (c) 2005 Elsevier Inc. All rights reserved.