On characteristics of local primitive matrices and digraphs

For local primitive $n$-vertex digraphs and matrices of order $n$, the following new characteristics are introduced: a matex is defined as a matrix $(\gamma_{i,j})$ of order $n$, where $\gamma_{i,j}=(i,j)-\exp\Gamma$, $1\leq i,j\leq n$; $k,r$-exporadius $\operatorname{exrd}_{k,r}\Gamma$ is defined as $\min_{I\times J\colon|I|=k,\ |J|=r}\gamma_{I,J}$, where $\gamma_{I,J}=\max_{(i,j)\in I\times J}\gamma_{i,j}$; $k,r$-expocenter is defined as a set $I\times J$, where $|I|=k$, $|J|=r$, $\gamma_{I,J}=\operatorname{exrd}_{k,r}\Gamma$. An approach to build the perfect $s$-boxes of order $k\times r$ using introduced characteristics is proposed. This approach is based on iterations of $n$-dimensional Boolean vectors set transformations with $n>\max(k,r)$. An exemplification of the function construction for perfect $s$-boxes of order $k\times r$ is presented.