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 G under component-wise group operation. Two-level group codes over the class of generalized quaternion groups, Q2m, TM > 3, are constructed using a binary code and a code over Z2m-i, the ring of integers modulo 2 , as component codes and a mapping / from Z2 X Z2m_i to Q2m. A set of necessary and sufficient conditions on the component codes is derived which will give group codes over Q2m. 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 Q2, it is shown that the Euclidean space codes obtained from the group codes over Q2m have Euclidean distance profiles which are independent of the coset representative selection involved in /. A closed-form expression for the minimum Euclidean distance of the resulting group codes over Q2m is obtained in terms of the Euclidean distances of the component codes. Finally, it is shown that all four-dimensional signal sets matched to Q2m have the same Euclidean distance profile and hence the Euclidean space codes corresponding to each signal set for a given group code over Q2m are automorphic Euclidean-distance equivalent.