  Recent experimental advances in Floquet engineering and controlling
dissipation in open systems have brought about unprecedented flexibility in
tailoring novel phenomena without any static and Hermitian analogues. It can be
epitomized by the various Floquet and non-Hermitian topological phases.
Topological classifications of either static/Floquet Hermitian or static
non-Hermitian systems based on the underlying symmetries have been well
established in the past several years. However, a coherent understanding and
classification of Floquet non-Hermitian (FNH) topological phases have not been
achieved yet. Here we systematically classify FNH topological bands for 54-fold
generalized Bernard-LeClair (GBL)symmetry classes and arbitrary spatial
dimensions using $K$-theory. The classification distinguishes two different
scenarios of the Floquet operator's spectrum gaps [dubbed as Floquet operator
(FO) angle-gapped and FO angle-gapless]. The results culminate into two
periodic tables, each containing 54-fold GBL symmetry classes. Our scheme
reveals FNH topological phases without any static/Floquet Hermitian and static
non-Hermitian counterparts. And our results naturally produce the periodic
tables of Floquet Hermitian topological insulators and Floquet unitaries. The
framework can also be applied to characterize the topological phases of bosonic
systems. We provide concrete examples of one and
two-dimensionalfermionic/bosonic systems. And we elucidate the meaning of the
topological invariants and their physical consequences. Our paper lays the
foundation for a comprehensive exploration of FNH topological bands. And it
opens a broad avenue toward uncovering unique phenomena and functionalities
emerging from the synthesis of periodic driving, non-Hermiticity, and band
topology.
