Quasi-cyclic code[1]

Description

A code \(C\) of length \(n\) over an alphabet is quasi-cyclic if, for each codeword \(c_1 \cdots c_{\ell} c_{\ell+1} \cdots c_n\), the string \(c_{n-\ell+1} \cdots c_n c_1 \cdots c_{n-\ell}\) where each entry is cyclically shifted by \(\ell\) increments is also a codeword.

The generator of an \([mn_0,mk_0]\) quasi-cyclic linear code is representable as a block matrix of \(m \times m\) circulant matrices [2].

Notes

A database of quasi-cyclic codes with searchable parameters such as block length and dimension is constructed and displayed here.

Parent

Children

  • Cyclic code — Quasi-cyclic codes with \(\ell=1\) are cyclic.
  • Skew-cyclic code — Under certain conditions, there is an equivalent quasi-cyclic code for every skew-cyclic code [3].

References

[1]
R. Townsend and E. Weldon, “Self-orthogonal quasi-cyclic codes”, IEEE Transactions on Information Theory 13, 183 (1967). DOI
[2]
Thomas A. Gulliver, Construction of quasi-cyclic codes, Thesis, University of New Brunswick, 1989.
[3]
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 information

Internal code ID: quasi_cyclic

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Zoo Code ID: quasi_cyclic

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

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

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