# 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

- Computer Science, Mathematics
- 2006

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

- Computer Science
- 2002

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

- Computer ScienceIEEE 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

- MathematicsIEEE 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

- Mathematics, Computer ScienceAdv. Math. Commun.
- 2008

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.)

- Computer ScienceIEEE Trans. Inf. Theory
- 1975

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

- Computer ScienceArXiv
- 2004

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

- Computer ScienceApplicable Algebra in Engineering, Communication and Computing
- 2006

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

- Computer Science, MathematicsIEEE Transactions on Information Theory
- 2006

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

- Computer Science, MathematicsIEEE Trans. Inf. Theory
- 1996

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.