Mapping design of a four-dimensional 32-ary signal constellation over an AWGN channel

In the bit-interleaved coded modulation with iterative decoding, it is desired that any two Hamming neighbors, i.e., signal constellation points whose binary labels are at unit Hamming distance, be separated by a large Euclidean distance. This paper determines the mappings for the 32-ary four-dimensional generalized cross constellation (32-4D-GCC) so that the minimum squared Euclidean distance d2min between Hamming neighbors is maximized. Among such mappings, those with minimum multiplicity N(d2min) are selected. To reduce the large search space, a set of “mapping templates,” each producing a collection of mappings with the same set partitions of binary labels, is introduced. Via enumeration of mapping templates, it is shown that the optimum d2min=16 and the optimum N(d2min)=16. Among thousands optimum mappings found by computer search, two of the best performance are presented.