Asymptotics for the Complexity of Boolean Functions with Small Number of Ones

The class $F_{n,k}$ of Boolean functions consisting of all functions of $n$ variables each of which outputs $1$ at exactly $k$ $n$-tuples of values of the variables is considered. For small $k$, for example, for $k<\ln n$, an asymptotics for the complexity of implementation of every function in $F_{n,k}$ by a circuit of functional elements in an irredundant basis containing $x\to y$ and $\overline{x\to y}$ is found.