Performance Variation of Gray Codes for Cropped Gaussian 16PAM Constellations

There are many ways to label symbols in a constellation. We investigate the performance impact of the Gray code choice both in the case of uniformly-spaced symbol constellations and cropped Gaussian constellations, with the BICM (bit-interleaved coded modulation) mutual information as a performance measure. We exhaustively search and find all 131 Gray codes, including cyclic and acyclic Gray codes, each representing a class of codes resulting in the same performance, that can be used with a symmetric 16PAM. Our results show that Gray codes having a bit position with only a single transition tend to outper-form codes where all bit positions have multiple transitions. The ranking of Gray code performance was similar when using both square PAM and optimized cropped Gaussian PAM, with only minor differences in the rankings of Gray codes. This indicates that joint selection of the cropped Gaussian constellation and the Gray code is not significantly important in designing a system. These results in the present paper can be applied to 256-symbol QAM constellations with in-phase/quadrature symmetry.