We present a correction to the above-named work. In Table III,$d_{free}^{2}$for new codes with$L=32$should be 3.561 (instead of 3.667) and 3.233 (instead of 3.385) for TCM-DE and DTCM, respectively. The corresponding paragraph on the same page, “For$L=32$, the best$d'^{2}_{free}$is …… in simulations.” should be modified as “For$L=32$, the best$d'^{2}_{free}$is$5\Delta _{1}^{2}+2\Delta _{0}^{2}$(for instance, two paths (0,0,0,0,0,0) and (0,2,0,1,2,2), so the resulting minimum squared distance is$\min [\Delta _{2}^{2},d'^{2}_{free}=3.233]=3.233$for 8PSK. There are some codes that have the best$d'^{2}_{free}$and we choose (C0,C2),(C3,C1),(C3,C1),(C2,C0),(C2,C0),(C1,C3) and (C1,C3) for$\sigma ^{(1)},\cdots ,\sigma ^{(7)}$in simulations.”. In addition, the trellis diagram in Fig. 12(b) should be corrected to Fig. 1.

Differential encoding (DE) is a classical technique at the transmitter that allows simple noncoherent detection at the receiver. On the other hand, trellis-coded modulation (TCM) is a bandwidth efficient technique which offers reliable data transmission. In this paper, we aim to find the best concatenation order of TCM and DE for channel phase coherence over N=2 symbols. Besides the well-known TCM followed by DE (called TCM-DE) , we propose a new trellis coding extension of DE which extends our earlier work, called differential trellis coded modulation (DTCM) . DTCM is TCM with DE defined in states where distinct states may have distinct DE functions. We propose design methods of DE functions for noncoherently non-catastrophic DTCM. For both TCM-DE and DTCM, we propose additive distance measures and set partitioning. Based on the proposed set partitioning, trellis codes are designed or searched for both TCM-DE and DTCM. Both minimum distances and simulation results show that TCM-DE outperforms DTCM and the obtained new codes are better than the original codes.