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.Notes
See books [1] 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
- Niemeier lattice— Extremal Type II self-dual codes of length 24 over \(\mathbb{Z}_6\) yield Niemeier lattices [2].
- Unimodular lattice— There are parallels between self-dual codes over \(\mathbb{Z}_{2k}\) and even unimodular lattices [3,4]. Type I (type II) codes over \(\mathbb{Z}_4\) yield type I (type II) lattices under Construction \(A_4\) [5; Thm. 12.5.12].
- \([11,6,5]_3\) Ternary Golay code— The extended ternary Golay code is self-dual.
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Primary Hierarchy
Parents
Self-dual code over \(\mathbb{Z}_q\)
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References
- [1]
- Self-Dual Codes and Invariant Theory (Springer-Verlag, 2006) DOI
- [2]
- M. Harada and M. Kitazume, “Z6-Code Constructions of the Leech Lattice and the Niemeier Lattices”, European Journal of Combinatorics 23, 573 (2002) DOI
- [3]
- 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
- [4]
- 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
- [5]
- W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, 2003) 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