Constellation labeling for linear encoders

This paper investigates optimal constellation labeling in the context of th e edge profile. A constellation's edge profile lists the minimum-distance e dge for each binary symbol error. The paper introduces the symmetric-ultrac omposite (SU) labeling structure and shows that this structure provides und ominated edge profiles for 2(n)-PSK, 2(n)-PAM, and 2(2n)-point square QAM. The SU structure is a generalization of the commonly used reflected binary Gray code. With the proper choice of basis vectors, SU labeling can support either set-partition or Gray-code labeling of 2n-PSK, 2(n)-PAM, and 2(2n)- point square QAM. Notably, there are Gray-code and set-partition labelings that do not have the SU structure. These labelings yield inferior edge prof iles. The SU structure does not apply to cross constellations. However, for any s tandard cross constellation with 32 or more points, a quasi-SU labeling str ucture can approximate the SU structure. With the correct choice of basis, quasi-SU labelings produce quasi-Gray labelings. However, the quasi-SU stru cture cannot support set-partition labeling. In fact, the quasi-SU structur e provides a better edge profile than standard set-partition labeling. Thus , for cross constellations there is a choice between edge profile optimalit y and the group structure provided by set-partitioning. Here, the correct c hoice depends on whether the encoder trellis has parallel branches.