The number of antichains in ranked partially ordered sets

We obtain the asymptotic behavior of the number of antichains in partially ordered sets whose diagrams are bipartite graphs that possess extension properties and whose number of vertices does not exceed $c\log_2\kappa$, where $\kappa$ is the minimum of the degrees of the vertices and $c$ is a constant less than 3. As a consequence we obtain the well-known asymptotic behavior of the number of binary codes with distance 2.