03-05
On Cyclic Convolutional Codes
by Heide Gluesing-Luerssen; Wiland Schmale
Preprint series: 03-05, Preprints
- MSC:
- 94B10 Convolutional codes
- 94B15 Cyclic codes
- 16S36 Ordinary and skew polynomial rings and semigroup rings, See also {20M25}
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
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