Generalising the concept of a complete permutation polynomial over a finite field, we define completness to level $k$ for $k\ge1$ in fields of odd characteristic. We construct two families of polynomials that satisfy the condition of high level completeness for all finite fields, and two more families complete to the maximum level a possible for large collection of finite fields. Under the binary operation of composition of functions one family of polynomials is an abelian group isomorphic to the additive group, while the other is isomorphic to the multiplicative group.

翻译：在有限域上推广完全置换多项式的概念，我们针对奇特征域定义了$k\ge1$级完备性。我们构造了两类多项式族，它们在所有有限域中满足高阶完备性条件；另两类多项式族则在大量有限域中达到可能的最高完备级。在函数的复合运算下，其中一个多项式族构成同构于加法群的阿贝尔群，另一个则同构于乘法群。