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Cyclic code[1–5]

Description

A code of length \(n\) over an alphabet 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.

Cousins

  • Lattice-shell code— Lattice-shell codewords are often permutations of a particular set of reference vectors, meaning that a cyclic permutation of a codeword yields another codeword.
  • Cyclic quantum code

Member of code lists

Primary Hierarchy

Classical Domain
Error-correcting code (ECC)
Block code
Quasi-twisted code
Quasi-cyclic code
Parents
Quasi-cyclic code
Quasi-cyclic codes with \(\ell=1\) are cyclic.
Constacyclic codeECC
Constacyclic codes with \(\alpha=1\) are cyclic.
Skew-cyclic codeECC
Skew-cyclic codes with \(\theta\) trivial are cyclic.
Group-orbit codeECC
All codewords of a cyclic code can be obtained from any codeword via cyclic shifts, meaning that the code consists of only one orbit.
Cyclic code
Children
Cyclic linear \(q\)-ary codeQR
Octacode
The octacode is a cyclic code over \(\mathbb{Z}_4\) with generator polynomial \(x^3+3x^2+2x+3\) extended by a parity check [6].

References

[1]
E. Prange, Cyclic Error-Correcting Codes in Two Symbols, TN-57-/03, (September 1957)
[2]
E. Prange, Some cyclic error-correcting codes with simple decoding algorithms, TN-58-156, (April 1958)
[3]
E. Prange, The use of coset equivalence in the analysis and decoding of group codes, TN-59-/64, (1959)
[4]
E. Prange, An algorithm for factoring xn - I over a finite field. TN-59-/75, (October 1959)
[5]
W. W. Peterson and E. J. Weldon, Error-correcting codes. MIT press 1972.
[6]
Self-Dual Codes and Invariant Theory (Springer-Verlag, 2006) DOI
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Zoo Code ID: cyclic

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

Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/properties/block/symmetry/cyclic/cyclic.yml.