The concept of a general nonlinear parity-check equation was first proposed by Pietrobon et al. who used them to construct rotationally invariant trellis codes for two dimensional signal sets. This paper applies this technique to trellis-coded modulation (TCM) codes with multidimensional MPSK and MQAM constellations to construct nonlinear phase rotationally invariant multi-D codes. Based on this construction technique, a series of the best fully phase rotationally invariant multi-D trellis codes for a variety of signal constellations is obtained through a systematic code search program. The code search results show that most codes achieve the same asymptotic coding gain as the best linear, but nonphase rotationally invariant codes found by Pietrobon et al.
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