Simplified Reed-Muller expressions for residue threshold functions

Residue threshold functions are a broad class of Boolean functions which includes all the unit-weighted threshold functions. In this paper we investigate the complexity of the Reed-Muller (RM) expressions for these functions. We prove that an important subclass of them has very simple RM expansions and determines the conditions that define it. As an interesting practical application, we show that the output functions of parallel counters belong to this subclass.