Let
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
$f(X)=\frac{a-b}{n}\left(\sum_{j=1}^{n-1}X^{j(q-1)/n}\right)X+\frac{a+b(n-1)}{n}X\in\mathbb{F}_qX]$
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.