Convolutional Codes Over Rings

Convolutional codes over rings behave quite differently from convolutional codes over fields, but they are the ones best suited for phase modulation. This behavior depends strongly on the structure of the underlying ring. Some basic concepts about ring convolutional codes, in particular over Zpr , and their structural properties such as basicity, systematicity, non-catastrophicity and minimality, is presented. Moreover, we describe a technique of constructing ring convolutional codes from linear block codes. Given a linear block code over the Galois ring GR(4,m) with a k×n generator matrix and minimum Hamming distance d, a rate-k/n convolutional code over the ring Z4 with memory at most m − 1 and squared Euclidean free distance at least 2d is constructed.