Functions without short implicents.Part I: lower estimates of weights

The paper is concerned with $n$-place Boolean functions not admitting implicents of $k$ variables, $1\le k<n$. Estimates for the minimal possible weight $w\left( {n,\;k} \right)$ of such functions are obtained. It is shown that $w\left( {n,\;1} \right) = 2$, $n = 2,3,...$, and $w\left( {n,\;2} \right)\sim{\log _2}n$ as $n \to \infty$, and moreover, for $k > 2$ there exists ${n_0}$ such that $w\left( {n,\;k} \right) > {2^{k - 2}} \cdot {\log _2}n$ for all $n > {n_0}$.