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×N points (where N=2ⁿ). The method, which was recently proposed by Wootters and co-workers [Gibbons et al., Phys. Rev. A 70, 062101 (2004)], is based on the use of the elements of the finite field GF(2ⁿ) 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.