  Following the method of induced group representations of Wigner-Mackay, the
explicit construction of all the unitary irreducible representations of the
discrete finite Heisenberg-Weyl group $HW_{2^s}$ over the discrete phase space
lattice $Z_{2^s}$ $\otimes$ $Z_{2^s}$ is presented. We explicitly determine
their characters and their fusion rules. We discuss possible physical
applications for finite quantum mechanics and quantum computation.
