On binary codes with distances d and d+2

We consider the problem of finding A₂ (n, \d₁, d₂\) defined as the maximal size of a binary (non-linear) code of length n with two distances d₁ and d₂. Binary codes with distances d and d+2 of size ²d{2 (d2+1) } can be obtained from 2-packings of an n-element set by blocks of cardinality d2+1. This value is far from the upper bound A₂ (n, \d₁, d₂\) 1+n2 proved recently by Barg et al. In this paper we prove that for every fixed d (d even) there exists an integer N (d) such that for every n N (d) it holds A₂ (n, \d, d+2\) =D (n, d2+1, 2), or, in other words, optimal codes are isomorphic to constant weight codes. We prove also estimates on N (d) for d=4 and d=6.