Multi-state Canalyzing Functions over Finite Fields

TL;DR

This paper generalizes Boolean canalyzing functions to finite fields, characterizes their classes, derives cardinalities, and explores asymptotic behaviors for large parameters.

Contribution

It extends the concept of canalyzing functions to finite fields and provides a complete classification and enumeration of these functions.

Findings

Characterization of eight classes of multi-state canalyzing functions

Derivation of their cardinalities and combinatorial identities

Asymptotic analysis for large number of variables or field size

Abstract

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 $q\to\infty$.