Coset-wise affine functions and cycle types of complete mappings

2021/09/08 by Alexander Bors, Qiang Wang, Bors, Alexander +1 · 1 citation
Computer Science · Mathematics · #15A21 #20B05 #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Limits and Structures in Graph Theory #Number Theory (math.NT) #Primary 12E20. Secondary 11T06

paper · pdf · doi:10.48550/arxiv.2109.03922

openalex publication_date 2021/09/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Let K be a finite field of characteristic p. We study a certain class of functions K→ K that agree with an \mathbbF_p-affine function K→ K on each coset of a given additive subgroup W of K - we call them W-coset-wise \mathbbF_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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• Circular sorting, strong complete mappings and wreath product constructions

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