  We study the class of discrete Wigner functions proposed by Gibbons et al.
[Phys. Rev. A 70, 062101 (2004)] to describe quantum states using a discrete
phase-space based on finite fields. We find the extrema of such functions for
small Hilbert space dimensions, and present a quantum information application:
a construction of quantum random access codes. These are constructed using the
complete set of phase-space point operators to find encoding states and to
obtain the codes' average success rates for Hilbert space dimensions 2,3,4,5,7
and 8.
