The constructive method for synthesis of balanced $k$-valued algebraic threshold functions

The interest of studying of threshold functions multiple-valued logic exists thanks to simplicity of their tasks and easiness of counting, which consists of the sum of scalar product. This sum can reflect either it traditional counting area of modern PC or in perspective optical computers [3]. The completeness of the basis of the multiple-valued threshold functions is proved in the article [6, 5]. This fact gives an opportunity to use them for realization of any multiple-valued system.
There is a class of multiple-valued threshold functions which expands thanks to taking reduction of linear form according to module. Also this class becomes a new one of algebraical threshold functions (ATF). The modular operation saves the simplicity of counting threshold functions, but also it expands its functional opportunities. The constructive evidence of existing balanced functions from class ATF, which are not threshold functions - is important result of the article.