Skew-cyclic code[1]

Description

A classical code \(C\) of length \(n\) over an alphabet \(R\) is skew-cyclic if there exists an automorphism, \(\theta\), of \(R\), such that for each string \(c_1 c_2 \cdots c_n\in C\), the skew-cyclically shifted string \(\theta(c_n) \theta(c_1) \cdots \theta(c_{n-1})\in C\). We say that \(C\) is a \(\theta\)-cyclic code over \(R\).

Decoding

Only given for skew-BCH codes, adapted froom standard BCH codes.

Realizations

Not directly implemented, but BCH codes form a subclass, and are used in DVD, solid state drive storage, etc.

Notes

Computer algebra software is used to find most codes of this type. Ref. [1] gives several examples of codes, which have slightly improved minimum distance for some \((n,k)\) codes.

Parent

  • Quasi-cyclic code — Under certain conditions, there is an equivalent quasi-cyclic code for every skew-cyclic code [2].

Cousins

Zoo code information

Internal code ID: skew_cyclic

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Zoo Code ID: skew_cyclic

Cite as:
“Skew-cyclic code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/skew_cyclic
BibTeX:
@incollection{eczoo_skew_cyclic, title={Skew-cyclic code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/skew_cyclic} }
Permanent link:
https://errorcorrectionzoo.org/c/skew_cyclic

References

[1]
Delphine Boucher, Willi Geiselmann, and Félix Ulmer, “Skew-cyclic codes”. math/0604603
[2]
I. Siap et al., “Skew cyclic codes of arbitrary length”, International Journal of Information and Coding Theory 2, 10 (2011). DOI

Cite as:

“Skew-cyclic code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/skew_cyclic

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/classical/properties/cyclic/skew_cyclic.yml.