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

Description

For any linear code \(C\) over \(\mathbb{Z}_4\), the dual code is the set of quaternary strings that are orthogonal to the codewords of \(C\) under the standard inner product modulo \(4\).

Protection

The dual of a code \(C=(n,4^{k_1} 2^{k_2})_{\mathbb{Z}_4}\) is \(C^{\perp} = (n,4^{n-k_1-k_2} 2^{k_2})\), whose generator matrix can be written in terms of the standard form of \(C\) [1; Prop. 1.2].

Cousin

Member of code lists

Primary Hierarchy

Classical Domain
Ring Kingdom
Ring codeECC
\(q\)-ary code over \(\mathbb{Z}_q\)ECC
Linear code over \(\mathbb{Z}_q\)ECC
Linear code over \(\mathbb{Z}_4\)
Parents
Linear code over \(\mathbb{Z}_4\)
Dual code over \(\mathbb{Z}_q\)Linear code over \(\mathbb{Z}_q\) ECC
Dual code over \(\mathbb{Z}_4\)
Children
Self-dual code over \(\mathbb{Z}_4\)

References

[1]
Z. X. Wan, Quaternary Codes (WORLD SCIENTIFIC, 1997) DOI
[2]
A. R. Hammons, 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: dual_over_z4

Cite as:
“Dual code over \(\mathbb{Z}_4\)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/dual_over_z4
BibTeX:
@incollection{eczoo_dual_over_z4, title={Dual code over \(\mathbb{Z}_4\)}, booktitle={The Error Correction Zoo}, year={2026}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/dual_over_z4} }
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Permanent link:
https://errorcorrectionzoo.org/c/dual_over_z4

Cite as:

“Dual code over \(\mathbb{Z}_4\)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/dual_over_z4

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/rings/over_zq/over_z4/linear_over_z4/dual_over_z4.yml.