Cyclic linear \(q\)-ary code

Description

A \(q\)-ary code of length \(n\) is cyclic if, for each codeword \(c_1 c_2 \cdots c_n\), the cyclically shifted string \(c_n c_1 \cdots c_{n-1}\) is also a codeword. A cyclic code is called primitive when \(n=q^r-1\) for some \(r\geq 2\). A shortened cyclic code is obtained from a cyclic code by taking only codewords with the first \(j\) zero entries, and deleting those zeroes.

Protection

Shift bound [1] gives a lower bound on the distance of cyclic \(q\)-ary codes.

Decoding

Meggitt decoder [2].

Notes

See Ch. 7 of Ref. [3] for an exposition on cyclic codes.

Parents

Children

Cousins

References

[1]
J. van Lint and R. Wilson, “On the minimum distance of cyclic codes”, IEEE Transactions on Information Theory 32, 23 (1986). DOI
[2]
J. Meggitt, “Error correcting codes and their implementation for data transmission systems”, IEEE Transactions on Information Theory 7, 234 (1961). DOI
[3]
F. J. MacWilliams and N. J. A. Sloane. The theory of error correcting codes. Elsevier, 1977.
[4]
Yongsheng Tang et al., “New quantum codes from dual-containing cyclic codes over finite rings”. 1608.06674
[5]
M. A. Tsfasman and S. G. Vlăduţ, Algebraic-geometric Codes (Springer Netherlands, 1991). DOI
[6]
G. G. La Guardia, “New Quantum MDS Codes”, IEEE Transactions on Information Theory 57, 5551 (2011). DOI

Zoo code information

Internal code ID: q-ary_cyclic

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Zoo Code ID: q-ary_cyclic

Cite as:
“Cyclic linear \(q\)-ary code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/q-ary_cyclic
BibTeX:
@incollection{eczoo_q-ary_cyclic, title={Cyclic linear \(q\)-ary code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/q-ary_cyclic} }
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Cite as:

“Cyclic linear \(q\)-ary code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/q-ary_cyclic

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/classical/q-ary_digits/cyclic/q-ary_cyclic.yml.