  This paper applies probabilistic amplitude shaping (PAS) to a cyclic
redundancy check (CRC) aided trellis coded modulation (TCM) to achieve the
short-blocklength random coding union (RCU) bound. In the transmitter, the
equally likely message bits are first encoded by distribution matcher to
generate amplitude symbols with the desired distribution. The binary
representations of the distribution matcher outputs are then encoded by a CRC.
Finally, the CRC-encoded bits are encoded and modulated by Ungerboeck's TCM
scheme, which consists of a $\frac{k_0}{k_0+1}$ systematic tail-biting
convolutional code and a mapping function that maps coded bits to channel
signals with capacity-achieving distribution. This paper proves that, for the
proposed transmitter, the CRC bits have uniform distribution and that the
channel signals have symmetric distribution. In the receiver, the serial list
Viterbi decoding (S-LVD) is used to estimate the information bits. Simulation
results show that, for the proposed CRC-TCM-PAS system with 87 input bits and
65-67 8-AM coded output symbols, the decoding performance under additive white
Gaussian noise channel achieves the RCU bound with properly designed CRC and
convolutional codes.
