A matrix ring description for cyclic convolutional codes

@article{GluesingLuerssen2007AMR,
  title={A matrix ring description for cyclic convolutional codes},
  author={Heide Gluesing-Luerssen and Fai-Lung Tsang},
  journal={ArXiv},
  year={2007},
  volume={abs/0708.1343}
}
In this paper, we study convolutional codes with a specific cyclic structure. By definition, these codes are left ideals in a certain skew polynomial ring. Using that the skew polynomial ring is isomorphic to a matrix ring we can describe the algebraic parameters of the codes in a more accessible way. We show that the existence of such codes with given algebraic parameters can be reduced to the solvability of a modified rook problem. It is our strong belief that the rook problem is always… 

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References

SHOWING 1-10 OF 23 REFERENCES

On the algebraic parameters of convolutional codes with cyclic structure

In this paper convolutional codes with cyclic structure will be investigated. These codes can be understood as left principal ideals in a suitable skew-polynomial ring. It has been shown in [4] that

On Cyclic Convolutional Codes

This work investigates the notion of cyclicity for convolutional codes as it has been introduced by Piret and Roos and shows how basic code properties and a minimal generator matrix can be read off from these objects.

On the structure of convolutional and cyclic convolutional codes

  • Kees Roos
  • Computer Science
    IEEE Trans. Inf. Theory
  • 1979
It is shown that any convolutional code has a canonical direct decomposition into subcodes and that this decomposition leads in a natural way to a minimal encoder.

Structure and constructions of cyclic convolutional codes

  • P. Piret
  • Mathematics
    IEEE Trans. Inf. Theory
  • 1976
A canonical decomposition of a CCC into minimal ideals is given which illuminates the cyclic structure and a number of CCC's with large free distance are constructed.

Group convolutional codes

Some theorems due to Jategaonkar that describe skew polynomial rings in terms of matrices are used to characterize minimal $S_3$-convolutional codes over the field of five elements.

An algebraic construction of rate 1/v -ary codes; algebraic construction (Corresp.)

It is proved here that the length of a rate 1/\nu q -ary code with this property is at most q\nu, and a class of such codes with lengths greater than q-nu/3 is constructed.

A class of one-dimensional MDS convolutional codes

It will turn out that one-dimensional convolutional codes are cyclic if and only if the field element used in the generator matrix has order n, which can be regarded as a generalization of the block code case.

On Doubly-Cyclic Convolutional Codes

After constructing doubly-cyclic CC’s, basic properties are derived on the basis of which distance properties of Reed-Solomon convolutional codes are investigated and some of them are optimal or near optimal with respect to distance and performance.

Convolutional Goppa codes

Convolutional Goppa codes over algebraic curves are defined and their corresponding dual codes are constructed, obtaining in particular some maximum-distance separable (MDS) convolutional codes.

On behaviors and convolutional codes

This paper defines a convolutional code as the dual of a complete linear behavior in the sense of Willems (1979) and describes a set of generalized first-order descriptions for convolutionan codes using ideas from systems theory.