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arXiv · 2109.03922

Coset-wise affine functions and cycle types of complete mappings

Alexander Bors·Qiang Wang
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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 $p>2$.

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Alexander Bors, Qiang Wang. 2021-09-08. Coset-wise affine functions and cycle types of complete mappings. https://arxiv.org/abs/2109.03922

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