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Linear code over \(\mathbb{Z}_4\)

Description

A linear code encoding \(K\) states (codewords) in \(n\) coordinates over the ring \(\mathbb{Z}_4\) of integers modulo 4.

Notes

Code Database, including quasi-cyclic and quasi-twisted codes [1].

Cousins

  • Linear binary 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 \(C\) [2; Thm. 12.2.3].
  • 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 \(C\) [2; Thm. 12.2.3].
  • Construction \(A_4\) code— Every linear code over \(\mathbb{Z}_4\) yields a lattice under Construction \(A_4\) [2; Sec. 12.5.3].
  • Melas code— The even-weight subcode of the Melas code can be lifted to a code over \(\mathbb{Z}_4\) [3].
  • Hergert code— Hergert codes can be seen, via the Gray map, as linear codes over \(\mathbb{Z}_4\) [4,5].
  • Delsarte-Goethals (DG) code— DG codes can be seen, via the Gray map, as extended linear cyclic codes over \(\mathbb{Z}_4\) [4,5].

Member of code lists

Primary Hierarchy

Classical Domain
Ring Kingdom
Ring codeECC
\(q\)-ary code over \(\mathbb{Z}_q\)
\(q\)-ary additive code over \(\mathbb{Z}_q\)
Linear code over \(\mathbb{Z}_q\)ECC
Parents
Linear code over \(\mathbb{Z}_q\)
Linear code over \(\mathbb{Z}_4\)
Children
Harada-Kitazume code
Octacode
Pseudo Golay code
Quaternary Golay code
Quaternary RM (QRM) code

References

[1]
N. Aydin, Y. Lu, and V. R. Onta, “An Updated Database of \(\mathbb{Z}_4\) Codes”, (2022) arXiv:2208.06832
[2]
W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, 2003) DOI
[3]
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
[4]
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
[5]
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
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Zoo Code ID: quaternary_over_z4

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
BibTeX:
@incollection{eczoo_quaternary_over_z4, title={Linear code over \(\mathbb{Z}_4\)}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/quaternary_over_z4} }
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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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/rings/over_zq/over_z4/linear_over_z4/quaternary_over_z4.yml.