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On Cyclic Convolutional Codes

by Heide Gluesing-Luerssen, Wiland Schmale.

Series: 2003-05, Preprints

94B10 Convolutional codes
94B15 Cyclic codes
16S36 Ordinary and skew polynomial rings and semigroup rings

We investigate the notion of cyclicity for convolutional codes
as it has been introduced in a short series of papers in the seventies.
Codes of this type are finitely generated free modules over a polynomial
ring which can also be described as left ideals in a skew polynomial ring.
Extending a result of the seventies we show that these ideals are always principal.
This leads to the notion of a generator polynomial just like for cyclic block codes.
Similarly a control polynomial can be introduced by considering the right annihilator ideal.
We also show how basic code properties and a minimal generator matrix can be read off from these objects.
A close link between polynomial and vector description of the codes is provided by certain
circulant matrices.

Algebraic Convolutional Coding Theory, Cyclic Codes, Skew Polynomial Ring