## 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.

## Parents

- Quasi-cyclic code — Quasi-cyclic codes with \(\ell=1\) are cyclic.
- Constacyclic code — Constacyclic codes with \(\alpha=1\) are cyclic.
- Skew-cyclic code — Skew-cyclic codes with \(\theta\) trivial are cyclic.
- Group-orbit code — All codewords of a cyclic code can be obtained from any codeword via cyclic shifts, meaning that the code consists of only one orbit.

## Children

- Difference-set cyclic (DSC) code
- Cyclic linear \(q\)-ary code
- 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].

## 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
- Generalized homological-product qubit CSS code — Iterative tensor-product codes can be constructed out of CSS codes that in turn stem from cyclic codes [7].

## 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
- [7]
- B. Audoux and A. Couvreur, “On tensor products of CSS Codes”, (2018) arXiv:1512.07081

## Page edit log

- Victor V. Albert (2022-07-20) — most recent
- Victor V. Albert (2021-11-30)
- Nolan Coble (2021-11-28)

## Cite as:

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