Linear block codes over cyclic groups

The main building block for the construction of a geometrically uniform coded modulation scheme is a subgroup of G^I, where G is a group generating a low-dimensional signal constellation and I is an index set. In this paper we study the properties of these subgroups when G is cyclic. We exploit the fact that any cyclic group of q elements is isomorphic to the additive group of Z_q (the ring of integers modulo q) so that we can make use of concepts related to linearity. Our attention is focused mainly on indecomposable cyclic groups (i.e., of prime power order), since they are the elementary “building blocks” of any abelian group. In analogy with the usual construction of linear codes over fields, we define a generator matrix and a parity check matrix. Trellis construction and bounds on the minimum Euclidean distance are also investigated. Some examples of coded modulation schemes based on this theory are also exhibited, and their performance evaluated