Block-coded modulation using two-level group codes over generalized quaternion groups

A length n group code over a group G is a subgroup of Gⁿ under component-wise group operation. Two-level group codes over the class of generalized quaternion groups, Q(2^m), m⩾3, are constructed using a binary code and a code over Z(2^m-1 ), the ring of integers modulo 2^m-1 as component codes and a mapping f from Z₂×Z(2^m-1)to Q(2^m ). A set of necessary and sufficient conditions on the component codes is derived which will give group codes over Q(2^m). Given the generator matrices of the component codes, the computational effort involved in checking the necessary and sufficient conditions is discussed. Starting from a four-dimensional signal set matched to Q(2 ^m), it is shown that the Euclidean space codes obtained from the group codes over Q(2^m) have Euclidean distance profiles which are independent of the coset representative selection involved in f. A closed-form expression for the minimum Euclidean distance of the resulting group codes over Q(2^m) is obtained in terms of the Euclidean distances of the component codes. Finally, it is shown that all four-dimensional signal sets matched to Q(2^m) have the same Euclidean distance profile and hence the Euclidean space codes corresponding to each signal set for a given group code over Q(2^m ) are automorphic Euclidean-distance equivalent