On the algebraic fundamentals of convolutional encoders over groups

The majority of the convolutional encoders known in the technical literature are over algebraic fields. Recently, it has been shown that these encoders essentially make use of the additive group of those fields. We take this approach and define the elementary convolutional encoder (ECE) over abelian groups and point out their main properties that will serve as a reference to the definition of general machines. By use of the Schreier product, the general convolutional encoder (GCE) is defined. As a consequence, the ECE is a particular case of the GCE. The Schreier product can be properly exploited in the design of the encoder. As an example, we provide two results about the machine only by looking at the properties of this product