Asymptotic numbers of non-equivalent codes in three notions of equivalence

Abstract

Let be the set of all subspaces of . The following three groups act on : (i) the symmetric group on the coordinate positions of ; (ii) the group of monomial transformations of ; (iii) the group of semi-linear monomial transformations of . The orbits of under these actions are the equivalence classes of linear codes in under three notions of equivalence: permutation equivalence, monomial equivalence and equivalence. Let , , N n,q Γ denote the numbers of the orbits under the three actions on respectively. It was recently proved that as n → ∞, and . In this article, we show that , where q = pr and p is a prime.