We discuss some interesting aspects of quantum nets on phase phase spaces based on F2n and Z2n appropriate to a finite state quantum systems described by 2n dimensional complex Hilbert space. The geometric properties of the phase spaces in the two cases differ from each other con- siderably owing to the differences in the algebraic properties of F2n and Z2n. The notion of quantum nets was first introduced by Gibbons et al in the context of phase spaces based on F2n who also gave a construction thereof making use of mutually unbiased bases available in such dimensions. Here we present an entirely algebraic construction of the quantum nets of Gibbons et al without explicit recourse to mutually unbiased bases. We also present a detailed account of the phase spaces based on Z2n and discuss the properties of the associated quantum nets. Some intriguing features of the spectra of the phase point operators that emerge from the study of systems in 2, 4, 8 and 16 dimensions are highlighted.