  Probabilistic amplitude shaping (PAS) can flexibly vary the spectral
efficiency (SE) of fiber-optic systems. In this paper, we demonstrate the
application of PAS to bit-wise hard decision decoding (HDD) of product codes
(PCs) by finding the necessary conditions to select the PC component codes. We
show that PAS with PCs and HDD yields gains up to $2.7$ dB and SE improvement
up to approximately $1$ bit/channel use compared to using PCs with uniform
signaling and HDD. Furthermore, we employ the recently introduced iterative
bounded distance decoding with combined reliability of PCs to improve
performance of PAS with PCs and HDD.
