  Majorana edge modes are candidate elements of topological quantum computing.
In this work, we purpose a Floquet engineering approach to generate arbitrarily
many non-Hermitian Majorana zero and $\pi$ modes at the edges of a
one-dimensional topological superconductor. Focusing on a Kitaev chain with
periodically kicked superconducting pairings and gain/losses in the chemical
potential or nearest neighbor hopping terms, we found rich non-Hermitian
Floquet topological superconducting phases, which are originated from the
interplay between drivings and non-Hermitian effects. Each of the phases is
characterized by a pair of topological winding numbers, which can in principle
take arbitrarily large integer values thanks to the applied driving fields.
Under open boundary conditions, these winding numbers also predict the number
of degenerate Majorana edge modes with quasienergies zero and $\pi$. Our
findings thus expand the family of Floquet topological phases in non-Hermitian
settings, with potential applications in realizing environmentally robust
Floquet topological quantum computations.
