Permutations of finite fields for check digit systems

• Authors: Rasha Shaheen;Arne Winterhof
• Venue: Designs, Codes and Cryptography
• Year: 2010

Let q be a prime power. For a divisor n of q 驴 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)卤X and f(f(X))卤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.