Permutations of finite fields for check digit systems
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) = a-b/n (Sigma(n-1)(j-1)X(j(q-1)/n)) X + a+b(n-1)/nX is an element of Fq[X] such that the five (not necessarily different) polynomials f(X), f(X) +/- X and f(f(X)) +/- X are all permutation polynomials over 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.
finite fields check digit systems permutation polynomials orthomorphisms complete mappings cyclotomy