LINEAR BLOCK-CODES OVER CYCLIC GROUPS

The main building block for the construction of a geometrically unifor m coded modulation scheme is a subgroup of G(I), where G is a group ge nerating a low-dimensional signal constellation and I is an index set. In this paper we study the properties of these subgroups when G is cy clic. We exploit the fact that any cyclic group of q elements is isomo rphic to the additive group of Z(q) (the ring of integers module 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 powe r order), since they are the elementary ''building blocks'' of any abe lian group. In analogy with the usual construction of linear codes ove r fields, we define a generator matrix and a parity check matrix. Trel lis 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.