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We analyze and further develop a new method to represent the quantum state ofa 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 elementsof the finite field $GF(2^n)$ to label the phase space axes. We present aself--contained overview of the method, we give new insights on some of itsfeatures and we apply it to investigate problems which are of interest forquantum information theory: We analyze the phase space representation ofstabilizer states and quantum error correction codes and present a phase spacesolution to the so--called ``mean king problem''.Comment: 18 pages, 16 figures; typos fixed, some minor corrections, figures of  the circuits were change

Qubits in phase space: Wigner-function approach to quantum-error correction and the mean-king problem