Qualified residue difference sets with zero

We previously established that biquadratic qualified residue difference sets exist for primes p if and only if p = 16x2 + 1 and sextic qualified residue difference sets exist if and only if p = 108x2 + 1. For example such sets exist for the primes 17 and 109, respectively. In this paper we point out that if zero is counted as a residue then we can obtain further qualified residue difference sets for both the biquadratic and the sextic residues. We give two theorems which state precisely when such biquadratic and sextic residue sets exist and a further existence theorem for more general powers.