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2003-05

On Cyclic Convolutional Codes

by Heide Gluesing-Luerssen, Wiland Schmale.


Series: 2003-05, Preprints

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

Abstract:
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.

Keywords:
Algebraic Convolutional Coding Theory, Cyclic Codes, Skew Polynomial Ring