Quasi-cyclic code[1] 


A block code of length \(n\) 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].


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





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“Generalized tail-biting convolutional codes,” Ph.D. dissertation, Univ. of Massachusetts, Amherst, 1985.
Y. Levy and J. Costello, Jr., “An algebraic approach to constructing convolutional codes from quasi-cyclic codes,” DIMACS Ser. Discr. Math. and Theor. Comp. Sci., vol. 14, pp. 189–198, 1993.
M. Esmaeili et al., “A link between quasi-cyclic codes and convolutional codes”, IEEE Transactions on Information Theory 44, 431 (1998) DOI
M. Hagiwara et al., “Spatially Coupled Quasi-Cyclic Quantum LDPC Codes”, (2011) arXiv:1102.3181
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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
  title={Quasi-cyclic code},
  booktitle={The Error Correction Zoo},
  editor={Albert, Victor V. and Faist, Philippe},
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“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/block/cyclic/quasi_cyclic.yml.