讲座内容：

We are interested in the value A_2(n, {d_1, d_2}) defined as the maximal size of a binary code of length n with two distances d_1 and d_2. It was recently proved by Barg et al. that A_2(n, {d_1, d_2}) ≤ 1 + n(n-1)/2. In this talk, we will discuss results from recent works and. One of these results is for the case d_2 = d_1 + 2, when d_1 is even. We prove that for every fixed even d there exists an integer n_0(d) such that for every n >= n_0(d) any optimal code of length n with distances d and d + 2 is isomorphic to a constant weight code (or equivalently, to a partial Steiner system). We also provide upper bounds on n_0(d) for d = 4 and d = 6. In addition, we provide linear on n upper bounds for the cases d_2>2d_1 and when d_1+d_2 is odd. The author was supported by the NSP P. Beron project CP-MACT. This talk is based on joint papers with Ivan Landjev and Assia Rousseva.