  We study an analytically solvable and experimentally relevant
number-conserving periodically driven $p$-wave superconductor. Such a system is
found to support generalized Majorana zero and $\pi$ modes which, despite being
non-Hermitian, are still capable of encoding qubits. Moreover, appropriate
winding numbers characterizing the topology of such generalized Majorana modes
are defined and explicitly calculated. We further discuss the fate of the
obtained generalized Majorana modes in the presence of finite charging energy.
Finally, we shed light on the quantum computing prospects of such modes by
demonstrating the robustness of their encoded qubits and explicitly braiding a
pair of generalized Majorana modes.
