On a method for constructing low-weight Boolean functions without majorants of the given number of variables

The problem of constructing Boolean functions without majorants of $k$ variables is reduced to the construction of a set $M$ of Boolean functions of $k-1$ variables such that for any different vectors $\overline\beta_1,\dots,\overline\beta_k\in V_{k-1}$ and for any $\alpha_1,\dots,\alpha_k\in\{0,1\}$ there exists a function $f\in M\colon f(\overline\beta_1)=\alpha_1,\dots,f(\overline\beta_k)=\alpha_k$. This approach permits to construct functions f of small weight having no $k-1$ variable majorants. Several families of such Boolean functions $f$ are constructed.