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.Protection
Extremal Type-II self-dual codes over \(\mathbb{Z}_4\) have been classified for \(n\leq 16\) [1,2], and there are 4744 such codes at \(n=24\) [3].Notes
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
- \(\Lambda_{24}\) Leech lattice— Each 4-frame of the Leech lattice corresponds to an extremal Type II self-dual code over \(\mathbb{Z}_4\) [3].
- Unimodular lattice— There are parallels between self-dual codes over \(\mathbb{Z}_{2k}\) and even unimodular lattices [4,5]. Type I (type II) codes over \(\mathbb{Z}_4\) yield type I (type II) lattices under Construction \(A_4\) [6; Thm. 12.5.12].
- Niemeier lattice— Extremal Type II self-dual codes of length 24 over \(\mathbb{Z}_6\) yield Niemeier lattices [7].
Member of code lists
Primary Hierarchy
Parents
Self-dual code over \(\mathbb{Z}_q\)
Children
Harada-Kitazume codes are extremal Type II self-dual codes over \(\mathbb{Z}_4\) [8].
The octacode is self-dual over \(\mathbb{Z}_4\).
Pseudo Golay codes are extremal Type II self-dual codes over \(\mathbb{Z}_4\) [9; Thm. 9].
References
- [1]
- 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
- [2]
- 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
- [3]
- 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
- [4]
- 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
- [5]
- 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
- [6]
- W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, 2003) DOI
- [7]
- M. Harada and M. Kitazume, “Z6-Code Constructions of the Leech Lattice and the Niemeier Lattices”, European Journal of Combinatorics 23, 573 (2002) DOI
- [8]
- 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
- [9]
- E. Rains, “Optimal self-dual codes over Z4”, Discrete Mathematics 203, 215 (1999) DOI
Page edit log
- Victor V. Albert (2024-04-29) — most recent
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