Description
A code encoding \(K\) states (codewords) in \(n\) coordinates over the ring \(\mathbb{Z}_4\) of integers modulo four that is an additive group. More technically, linear codes over \(\mathbb{Z}_4\) are submodules of \(\mathbb{Z}_4^n\).Notes
Code Database, including quasi-cyclic and quasi-twisted codes [1].See books [2–4] for introductions.Cousins
- Construction \(A_4\) code— Every linear code over \(\mathbb{Z}_4\) yields a lattice under Construction \(A_4\) [3; Sec. 12.5.3].
- Melas code— The even-weight subcode of the Melas code can be lifted to a code over \(\mathbb{Z}_4\) [5].
- Hergert code— Hergert codes can be seen, via the Gray map, as linear codes over \(\mathbb{Z}_4\) [6,7].
- Gray code— A linear code \(C\) over \(\mathbb{Z}_4\) can be mapped, via the Gray map, to a binary code. The binary code is linear if and only if doubling the component-wise product of any two codewords in \(C\) yields another codeword in \(C\) [3; Thm. 12.2.3].
- Delsarte-Goethals (DG) code— DG codes can be seen, via the Gray map, as extended linear cyclic codes over \(\mathbb{Z}_4\) [6,7].
Member of code lists
Primary Hierarchy
Parents
Linear binary codes are linear \(q\)-ary codes over \(\mathbb{Z}_q\) for \(q=4\).
Linear code over \(\mathbb{Z}_4\)
Children
The quaternary Golay code is an extremal Type II self-dual code over \(\mathbb{Z}_4\) by virtue of its parameters [8].
References
- [1]
- N. Aydin, Y. Lu, and V. R. Onta, “An Updated Database of \(\mathbb{Z}_4\) Codes”, (2022) arXiv:2208.06832
- [2]
- Z. X. Wan, Quaternary Codes (WORLD SCIENTIFIC, 1997) DOI
- [3]
- W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, 2003) DOI
- [4]
- R. Roth, Introduction to Coding Theory (Cambridge University Press, 2006) DOI
- [5]
- A. Alahmadi, H. Alhazmi, T. Helleseth, R. Hijazi, N. Muthana, and P. Solé, “On the lifted Melas code”, Cryptography and Communications 8, 7 (2015) DOI
- [6]
- A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Sole, “The Z/sub 4/-linearity of Kerdock, Preparata, Goethals, and related codes”, IEEE Transactions on Information Theory 40, 301 (1994) DOI
- [7]
- A. R. Hammons Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Solé, “The Z_4-Linearity of Kerdock, Preparata, Goethals and Related Codes”, (2002) arXiv:math/0207208
- [8]
- A. Munemasa and R. A. L. Betty, “Classification of extremal type II \(\)\mathbb {Z}_4\(\)-codes of length 24”, Designs, Codes and Cryptography 92, 771 (2023) DOI
Page edit log
- Victor V. Albert (2022-03-04) — most recent
Cite as:
“Linear code over \(\mathbb{Z}_4\)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quaternary_over_z4