  The original Wigner function provides a way of representing in phase space
the quantum states of systems with continuous degrees of freedom. Wigner
functions have also been developed for discrete quantum systems, one popular
version being defined on a 2N x 2N discrete phase space for a system with N
orthogonal states. Here we investigate an alternative class of discrete Wigner
functions, in which the field of real numbers that labels the axes of
continuous phase space is replaced by a finite field having N elements. There
exists such a field if and only if N is a power of a prime; so our formulation
can be applied directly only to systems for which the state-space dimension
takes such a value. Though this condition may seem limiting, we note that any
quantum computer based on qubits meets the condition and can thus be
accommodated within our scheme. The geometry of our N x N phase space also
leads naturally to a method of constructing a complete set of N+1 mutually
unbiased bases for the state space.
