  We analyze and further develop a new method to represent the quantum state of
a system of $n$ qubits in a phase space grid of $N\times N$ points (where
$N=2^n$). The method, which was recently proposed by Wootters and co--workers
(Gibbons {\it et al.}, quant-ph/0401155), is based on the use of the elements
of the finite field $GF(2^n)$ to label the phase space axes. We present a
self--contained overview of the method, we give new insights on some of its
features and we apply it to investigate problems which are of interest for
quantum information theory: We analyze the phase space representation of
stabilizer states and quantum error correction codes and present a phase space
solution to the so--called ``mean king problem''.
