Let \(r\ge 3\) 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 \(f=\tau \circ \sigma _M\circ \tau ^{-1}\) where \(\tau \) is a PP over an extension field \({\mathbb {F}}_{q^d}\) and \(\sigma _M\) is an invertible linear map over \({\mathbb {F}}_{q^d}\). When \(\tau \) is additive, we give a general construction of r-regular CPPs for any positive integer r. When \(\tau \) 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 et al. (Des Codes Cryptogr 90:545–575, 2022).

令\(r\ge 3\)为正整数，\({\mathbb {F}}_q\)为具有q个元素的有限域。在本文中，我们考虑形式为\(f=\tau \circ \sigma _M\circ \tau ^{-1}\) 的映射的r正则完全置换性质，其中\(\tau \)是扩展域\({\mathbb {F}}_{q^d}\)上的并程，而\(\sigma _M\)是 \({\mathbb {F}}_{q^d} 上的可逆线性映射\)。当\(\tau \)是可加性时，我们给出任意正整数r 的r正则 CPP的一般构造。当\(\tau \)不是相加的，我们在\(r=3,4,5,6,7\)和任意奇数正整数r的扩展域上给出了许多常规 CPP 的例子。这些例子是Xu 等人构建的第一类r正则 CPP的推广。（Des Codes Cryptogr 90：545–575，2022）。