Constellation labeling for linear encoders

This paper investigates optimal constellation labeling in the context of the edge profile. A constellation´s edge profile lists the minimum-distance edge for each binary symbol error. The paper introduces the symmetric-ultracomposite (SU) labeling structure and shows that this structure provides undominated edge profiles for 2ⁿ-PSK, 2 ⁿ-PAM, and 2²ⁿ-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, 2ⁿ-PAM, and 2²ⁿ-point square QAM. Notably, there are Gray-code and set-partition labelings that do not have the SU structure. These labelings yield inferior edge profiles. The SU structure does not apply to cross constellations. However, for any standard cross constellation with 32 or more points, a quasi-SU labeling structure can approximate the SU structure. With the correct choice of basis, quasi-SU labelings produce quasi-Gray labelings. However, the quasi-SU structure cannot support set-partition labeling. In fact, the quasi-SU structure provides a better edge profile than standard set-partition labeling. Thus, for cross constellations there is a choice between edge profile optimality and the group structure provided by set-partitioning. Here, the correct choice depends on whether the encoder trellis has parallel branches