# 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 \(R\).

## 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

## Child

- Cyclic code — Skew-cyclic codes with \(\theta\) trivial are cyclic.

## Cousins

- Quasi-cyclic code — Under certain conditions, there is an equivalent quasi-cyclic or cyclic code for every skew-cyclic code [2].
- Skew-cyclic CSS code — Skew-cyclic CSS codes are constructed from classical skew-cyclic codes over rings.

## References

- [1]
- D. Boucher, W. Geiselmann, and F. Ulmer, “Skew-cyclic codes”, (2006) arXiv:math/0604603
- [2]
- I. Siap et al., “Skew cyclic codes of arbitrary length”, International Journal of Information and Coding Theory 2, 10 (2011) DOI

## Page edit log

- Nolan Coble (2021-12-03) — most recent

## Cite as:

“Skew-cyclic code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/skew_cyclic