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

Abstract

We consider the problem of finding $A_2(n,\{d_1,d_2\})$ defined as the maximal size of a binary (non-linear) code of length $n$ with two distances $d_1$ and $d_2$. Binary codes with distances $d$ and $d+2$ of size $\sim\frac{n^2}{\frac{d}{2}(\frac{d}{2}+1)}$ can be obtained from $2$-packings of an $n$-element set by blocks of cardinality $\frac{d}{2}+1$. This value is far from the upper bound $A_2(n,\{d_1,d_2\})\le1+{n\choose2}$ 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 $n\ge N(d)$ it holds $A_2(n,\{d,d+2\})=D(n,\frac{d}{2}+1,2)$, or, in other words, optimal codes are isomorphic to constant weight codes. We prove also estimates on $N(d)$ for $d=4$ and $d=6$.