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
- Cyclic code
- Linear \(q\)-ary code
- Group code — A length-\(n\) cyclic \(q\)-ary linear code is an abelian group 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
- Galois-qudit CSS code — Galois CSS codes can be constructed using self-orthogonal \(q\)-ary cyclic codes [4].
- Generalized RM (GRM) code — GRM codes with nonzero evaluation points are cyclic ([5], pg. 52).
- Quantum maximum-distance-separable (MDS) code — Quantum MDS codes can be constructed from \(q\)-ary cyclic codes using the Hermitian construction [6].
- 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]
- 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
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