On binary codes with distances d and $d+2$

We consider the problem of finding A₂(n,₁,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 ~n²d/2(d/2+1) can be obtained from $2$-packings of an n-element set by blocks of cardinality d/2+1. This value is far from the upper bound A₂(n,₁,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+2\)=D(n,d/2+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 $d=6$.