Skew-cyclic code[1] 


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 \(R\).


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


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.



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


  • 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.


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

Cite as:
“Skew-cyclic code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021.
@incollection{eczoo_skew_cyclic, title={Skew-cyclic code}, booktitle={The Error Correction Zoo}, year={2021}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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Cite as:

“Skew-cyclic code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021.