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We introduce the multi-dimensional permutation-modulation format in coherent optical communication systems and analyze its performance, focusing on the power efficiency and the spectral efficiency. In the case of four-dimensional (4D) modulation, the polarization-switched quadrature phase-shift keying (PS-QPSK) modulation format and the polarization quadrature-amplitude modulation (POL-QAM) format can be classified into the permutation modulation format. Other than these well-known modulation formats, we find novel modulation formats trading-off between the power efficiency and the spectral efficiency. With the increase in the dimension, the spectral efficiency can more closely approach the channel capacity predicted from the Shannon's theory. We verify these theoretical characteristics through computer simulations of the symbol-error rate (SER) and bit-error rate (BER) performances. For example, the newly-found eight-dimensional (8D) permutation-modulation format can improve the spectral efficiency up to 2.75 bit/s/Hz/pol/channel, while the power penalty against QPSK is about 1 dB at BER=10(-3).

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We apply the eight-state trellis-coded modulation (TCM) using signal constellations of four-dimensional M-ary quadrature-amplitude modulation (4D-MQAM) to optical communication systems for the first time to our knowledge. In the TCM scheme, the free distance of the trellis diagram is equal to the minimum distance between constellation points in partitioned subsets, which enlarges the coding gain effectively. In fact, its asymptotic power efficiency is 3-dB larger than that of the set-partitioned 4D-MQAM (SP-4D-MQAM) format, while their spectral efficiencies are the same. Such theoretical predictions are confirmed through computer simulations on eight-state TCM with constellations of 4D-4QAM (i.e., 4D quadrature phase-shift keying: 4D-QPSK) and 4D-16QAM. In particular, eight-state TCM with 4D-QPSK constellations is practically important because of its simple encoder structure, relatively low computational cost, and high coding gain against dual-polarization QPSK (DP-QPSK) and SP-4D-QPSK. Through measurements of its bit-error rate (BER) performance, we confirm that the coding gain against DP-QPSK is about 3 dB at BER=10(-3).

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