  In this letter, for the discrete parity-time-symmetric nonlocal nonlinear
Schr\"{o}dinger equation, we construct the Darboux transformation, which
provides an algebraic iterative algorithm to obtain a series of analytic
solutions from a known one. To illustrate, the breathing-soliton solutions,
periodic-wave solutions and localized rational soliton solutions are derived
with the zero and plane-wave solutions as the seeds. The properties of those
solutions are also discussed, and particularly the asymptotic analysis reveals
all possible cases of the interaction between the discrete rational dark and
antidark solitons.
