  Some hadrons have the exotic quantum numbers that the traditional $\bar q q$
mesons and $qqq$ baryons can not reach, such as $J^{PC} =
0^{--}/0^{+-}/1^{-+}/2^{+-}/3^{-+}/4^{+-}$, etc. We investigate for the first
time the exotic quantum number $J^{PC}=4^{+-}$, and study the fully-strange
tetraquark states with such an exotic quantum number. We systematically
construct all the diquark-antidiquark interpolating currents, and apply the
method of QCD sum rules to calculate both the diagonal and off-diagonal
correlation functions. The obtained results are used to construct three mixing
currents that are nearly non-correlated, and we use one of them to extract the
mass of the lowest-lying state to be $2.85^{+0.19}_{-0.22}$ GeV. We apply the
Fierz rearrangement to transform this mixing current to be the combination of
three meson-meson currents, and the obtained Fierz identity suggests that this
state dominantly decays into the $P$-wave $\phi(1020) f_2^\prime(1525)$
channel. This fully-strange tetraquark state of $J^{PC}=4^{+-}$ is a purely
exotic hadron to be potentially observed in future particle experiments.
