Coset-wise affine functions and cycle types of complete mappings

TL;DR

This paper investigates a class of functions on finite fields that are affine on cosets, characterizes when they are complete mappings, and provides new examples of cycle types, including a permutation with a single cycle for certain fields.

Contribution

It introduces coset-wise affine functions, describes their permutation group structure, and characterizes complete mappings within this class, offering new cycle type examples.

Findings

Functions form an imprimitive wreath product permutation group.

Characterization of which functions are complete mappings.

Existence of complete mappings with a single cycle for fields with characteristic p>2.

Abstract

Let $K$ be a finite field of characteristic $p$. We study a certain class of functions $K\rightarrow K$ that agree with an $\mathbb{F}_p$-affine function $K\rightarrow K$ on each coset of a given additive subgroup $W$ of $K$ - we call them $W$-coset-wise $\mathbb{F}_p$-affine functions of $K$. We show that these functions form a permutation group on $K$ with the structure of an imprimitive wreath product and characterize which of them are complete mappings of $K$. As a consequence, we are able to provide various new examples of cycle types of complete mappings of $K$, including that $K$ has a complete mapping moving all elements of $K$ in one cycle if $p>2$.