The class of balanced algebraic threshold functions

The paper proposes an approach to the construction of a class of balanced algebraic threshold functions (ATF). The function $f$ of $k$-valued logic is called ATF if there are sequences $\mathbf c=(c_0,c_1,\dots,c_n)$, $\mathbf b=(b_0,b_1,\dots,b_k)$ of integers and the natural modulus $m$ such that $f(x_1,x_2,\dots,x_n)=\alpha\Leftrightarrow b_\alpha\leq(c_0+c_1x_1+c_2x_2+\dots+c_n x_n)\mod m<b_{\alpha+1}$ for any $\alpha\in\Omega_k=\{0,1,\dots,k-1\}$. The triple $(\mathbf c;\mathbf b;m)$ is called the structure of the function $f$. The central result of the paper is a class of balanced ATF constructed in the following way: if an ATF $f$ has a structure $(\mathbf c,\mathbf b,m)=((c_0,c_1,c_2,\dots,c_n);(0,p,2p,\dots,kp);kp)$ where $c_i=pq$ and $(q,k)=1$, then this function is balanced. Such functions can be used as coordinate functions of substitutions.