Multi-state Canalyzing Functions over Finite Fields

In this paper, we extend the definition of Boolean canalyzing functions to the canalyzing functions over finite field $\mathbb{F}_{q}$, where $q$ is a power of a prime. We obtain the characterization of all the eight classes of such functions as well as their cardinality. When $q=2$, we obtain a combinatorial identity by equating our result to the formula in \cite{Win}. Finally, for a better understanding to the magnitude, we obtain the asymptotes for all the eight cardinalities as either $n\to\infty$ or $q\to\infty$.