Permutations of finite fields for check digit systems

AbstractLet q be a prime power. For a divisor n of q − 1 we prove an asymptotic formula for the number of polynomials of the form $$f(X)=\frac{a-b}{n}\left(\sum_{j=1}^{n-1}X^{j(q-1)/n}\right)X+\frac{a+b(n-1)}{n}X\in\mathbb{F}_q[X]$$such that the five (not necessarily different) polynomials f(X), f(X)±X and f(f(X))±X are all permutation polynomials over $${\mathbb{F}_q}$$ . Such polynomials can be used to define check digit systems that detect the most frequent errors: single errors, adjacent transpositions, jump transpositions, twin errors and jump twin errors.