Block-coded PSK modulation using two-level group codes over dihedral groups

A length n group code over a group G is a subgroup of Gⁿ under component-wise group operation. Group codes over dihedral groups D_M, with 2M elements, that are two-level constructible using a binary code and a code over Z_M residue class integer ring modulo M, as component codes are studied for arbitrary M. A set of necessary and sufficient conditions on the component codes for the two-level construction to result in a group code over D_M are obtained. The conditions differ for M odd and even. Using two-level group codes over D_M as label codes, the performance of a block-coded modulation scheme is discussed under all possible matched labelings of 2M-APSK and 2M-SPSK (asymmetric and symmetric PSK) signal sets in terms of the minimum squared Euclidean distance. Matched labelings that lead to automorphic Euclidean distance equivalent codes are identified. It is shown that depending upon the ratio of Hamming distances of the component codes some labelings perform better than others. The best labeling is identified under a set of restrictive conditions. Finally, conditions on the component codes for phase rotational invariance properties of the signal space codes are discussed