[Jump to code hierarchy]

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

Description

An additive linear code \(C\) over \(\mathbb{Z}_q\) that is equal to its dual, \(C^\perp = C\), where the dual is defined with respect to the standard inner product.

Self-dual codes for \(q=4\) contain \(2^n\) 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].

Notes

See books [5] for more on self-dual codes over \(\mathbb{Z}_q\).See Database of self-dual codes by M. Harada and A. Munemasa for a database of self-dual codes over \(\mathbb{Z}_{4}\), \(\mathbb{Z}_{6}\), \(\mathbb{Z}_{8}\), \(\mathbb{Z}_{9}\), and \(\mathbb{Z}_{10}\).

Cousins

Primary Hierarchy

Parents
Self-dual code over \(\mathbb{Z}_q\)
Children
The \(C_{m,r}\) code is a Type IV self-dual code over \(\mathbb{Z}_4\) [11].
Harada-Kitazume codes are extremal Type II self-dual codes over \(\mathbb{Z}_4\) [12].
The Klemm code is a Type IV self-dual code over \(\mathbb{Z}_4\) [11].
The octacode is self-dual over \(\mathbb{Z}_4\).
Pseudo Golay codes are extremal Type II self-dual codes over \(\mathbb{Z}_4\) [13; Thm. 9].
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]
Self-Dual Codes and Invariant Theory (Springer-Verlag, 2006) DOI
[6]
A. Bonnecaze, P. Sole, C. Bachoc, and B. Mourrain, “Type II codes over Z/sub 4/”, IEEE Transactions on Information Theory 43, 969 (1997) DOI
[7]
E. Bannai, S. T. Dougherty, M. Harada, and M. Oura, “Type II codes, even unimodular lattices, and invariant rings”, IEEE Transactions on Information Theory 45, 1194 (1999) DOI
[8]
W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, 2003) DOI
[9]
M. Harada and M. Kitazume, “Z6-Code Constructions of the Leech Lattice and the Niemeier Lattices”, European Journal of Combinatorics 23, 573 (2002) DOI
[10]
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
[11]
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
[12]
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
[13]
E. Rains, “Optimal self-dual codes over Z4”, Discrete Mathematics 203, 215 (1999) DOI
Page edit log

Your contribution is welcome!

on github.com (edit & pull request)— see instructions

edit on this site

Zoo Code ID: self_dual_over_zq

Cite as:
“Self-dual code over \(\mathbb{Z}_q\)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/self_dual_over_zq
BibTeX:
@incollection{eczoo_self_dual_over_zq, title={Self-dual code over \(\mathbb{Z}_q\)}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/self_dual_over_zq} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:
https://errorcorrectionzoo.org/c/self_dual_over_zq

Cite as:

“Self-dual code over \(\mathbb{Z}_q\)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/self_dual_over_zq

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/rings/over_zq/dual/self_dual_over_zq.yml.