# 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.

## Parents

- Cyclic code
- Linear \(q\)-ary code
- Group-algebra code — A length-\(n\) cyclic \(q\)-ary linear code is an abelian group-algebra code for the cyclic group with \(n\) elements \( \mathbb{Z}_n \).

## Children

- Bose–Chaudhuri–Hocquenghem (BCH) code
- Dodecacode
- \(q\)-ary duadic code
- \(q\)-ary parity-check code — Since permutations preserve coordinate sums, the cyclic permutation of a parity-check codeword is another codeword.

## Cousins

- \(q\)-ary linear LTC — Cyclic linear codes cannot be \(c^3\)-LTCs [4]. Codeword symmetries are in general an obstruction to achieving such LTCs [5].
- Dual linear code — See Refs. [6][7] for tables of cyclic self-dual codes.
- Galois-qudit CSS code — Galois CSS codes can be constructed using self-orthogonal \(q\)-ary cyclic codes [8].
- Generalized RM (GRM) code — GRM codes with nonzero evaluation points are cyclic ([9], pg. 52).
- Quantum maximum-distance-separable (MDS) code — Quantum MDS codes can be constructed from \(q\)-ary cyclic codes using the Hermitian construction [10].
- Reed-Solomon (RS) code — If the length divides \(q-1\), then it is possible to construct a cyclic RS code.
- \(q\)-ary Hamming code — Hamming codes are equivalent to cyclic codes when \(q\) and \(r\) are relatively prime ([3], pg. 194).

## 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]
- L. Babai, A. Shpilka, and D. Stefankovic, “Locally Testable Cyclic Codes”, IEEE Transactions on Information Theory 51, 2849 (2005). DOI
- [5]
- M. Sudan, “Invariance in Property Testing”, Property Testing 211 (2010). DOI
- [6]
- Yan Jia, San Ling, and Chaoping Xing, “On Self-Dual Cyclic Codes Over Finite Fields”, IEEE Transactions on Information Theory 57, 2243 (2011). DOI
- [7]
- S. Jitman et al., “Abelian Codes in Principal Ideal Group Algebras”, IEEE Transactions on Information Theory 59, 3046 (2013). DOI
- [8]
- Yongsheng Tang et al., “New quantum codes from dual-containing cyclic codes over finite rings”. 1608.06674
- [9]
- M. A. Tsfasman and S. G. Vlăduţ, Algebraic-geometric Codes (Springer Netherlands, 1991). DOI
- [10]
- G. G. La Guardia, “New Quantum MDS Codes”, IEEE Transactions on Information Theory 57, 5551 (2011). DOI

## Page edit log

- Victor V. Albert (2022-07-13) — most recent

## Zoo code information

## 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