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Self-dual code over \(\mathbb{Z}_4\)

Description

An additive linear code \(C\) over \(\mathbb{Z}_4\) that is equal to its dual, \(C^\perp = C\), where the dual is defined with respect to the standard inner product. The code contains \(2^n\) codewords [1; Corr. 1.3].

Protection

Extremal Type-II self-dual codes over \(\mathbb{Z}_4\) have been classified for \(n\leq 16\) [2,3], and there are 4744 such codes at \(n=24\) [4].

Cousin

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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 \(\mathbb{Z}_4\)
Dual code over \(\mathbb{Z}_4\)Linear code over \(\mathbb{Z}_q\) ECC
Parents
Dual code over \(\mathbb{Z}_4\)
Self-dual code over \(\mathbb{Z}_q\)Linear code over \(\mathbb{Z}_q\) ECC
Self-dual code over \(\mathbb{Z}_4\)
Children
\(C_{m,r}\) code
The \(C_{m,r}\) code is a Type IV self-dual code over \(\mathbb{Z}_4\) [5].
Harada-Kitazume code
Harada-Kitazume codes are extremal Type II self-dual codes over \(\mathbb{Z}_4\) [6].
Klemm code
The Klemm code is a Type IV self-dual code over \(\mathbb{Z}_4\) [5].
Octacode
The octacode is self-dual over \(\mathbb{Z}_4\).
Pseudo Golay code
Pseudo Golay codes are extremal Type II self-dual codes over \(\mathbb{Z}_4\) [7; Thm. 9].
Extended quaternary Golay code
The extended quaternary Golay code is an extremal Type II self-dual code over \(\mathbb{Z}_4\) by virtue of its parameters [4].

References

[1]
Z. X. Wan, Quaternary Codes (WORLD SCIENTIFIC, 1997) DOI
[2]
J. Fields, P. Gaborit, J. S. Leon, and V. Pless, “All self-dual Z/sub 4/ codes of length 15 or less are known”, IEEE Transactions on Information Theory 44, 311 (1998) DOI
[3]
V. Pless, J. S. Leon, and J. Fields, “All Z4Codes of Type II and Length 16 Are Known”, Journal of Combinatorial Theory, Series A 78, 32 (1997) DOI
[4]
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
[5]
S. T. Dougherty, P. Gaborit, M. Harada, A. Munemasa, and P. Sole, “Type IV self-dual codes over rings”, IEEE Transactions on Information Theory 45, 2345 (1999) DOI
[6]
M. Harada and M. Kitazume, “Z4-Code Constructions for the Niemeier Lattices and their Embeddings in the Leech Lattice”, European Journal of Combinatorics 21, 473 (2000) DOI
[7]
E. Rains, “Optimal self-dual codes over Z4”, Discrete Mathematics 203, 215 (1999) DOI
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Zoo Code ID: self_dual_over_z4

Cite as:
“Self-dual code over \(\mathbb{Z}_4\)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/self_dual_over_z4
BibTeX:
@incollection{eczoo_self_dual_over_z4, title={Self-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/self_dual_over_z4} }
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Cite as:

“Self-dual code over \(\mathbb{Z}_4\)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/self_dual_over_z4

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/rings/over_zq/over_z4/linear_over_z4/self_dual/self_dual_over_z4.yml.