论文标题
常规完整置换多项式的一般结构
A general construction of regular complete permutation polynomials
论文作者
论文摘要
令$ r \ geq 3 $为正整数，$ \ mathbb {f} _q $带有$ q $元素的有限字段。在本文中，我们考虑了$ f $ r $的完整置换属性，其形式为$ f =τ\circσ_m\circtτ^{ - 1} $，其中$τ$是扩展字段$ \ mathbb {f} _ {q^d} $ y mathbb {f}的PP $ \ mathbb {f} _ {q^d} $。对于任何正整数$ r $，我们提供了$ r $ r $的PPS的一般结构。当$τ$是添加剂时，我们为任何正整数$ r $ $ $ $ r $ $ r $ regiargular cpps提供了一般结构。当$τ$不是添加剂时，我们为$ r = 3,4,5,6,7 $的扩展字段提供了许多常规CPP示例，并且对于任意的奇数正整数$ r $。这些示例是Xu，Zeng和Zhang构建的$ r $ regigular CPP的一类概括（des。CodesCryptogr。90，545-575（2022））。
Let $r\geq 3$ be a positive integer and $\mathbb{F}_q$ the finite field with $q$ elements. In this paper, we consider the $r$-regular complete permutation property of maps with the form $f=τ\circσ_M\circτ^{-1}$ where $τ$ is a PP over an extension field $\mathbb{F}_{q^d}$ and $σ_M$ is an invertible linear map over $\mathbb{F}_{q^d}$. We give a general construction of $r$-regular PPs for any positive integer $r$. When $τ$ is additive, we give a general construction of $r$-regular CPPs for any positive integer $r$. When $τ$ is not additive, we give many examples of regular CPPs over the extension fields for $r=3,4,5,6,7$ and for arbitrary odd positive integer $r$. These examples are the generalization of the first class of $r$-regular CPPs constructed by Xu, Zeng and Zhang (Des. Codes Cryptogr. 90, 545-575 (2022)).
