Description
The unique self-dual linear code of length 8 over \(\mathbb{Z}_4\) with generator matrix \begin{align} \begin{pmatrix} 3 & 3 & 2 & 3 & 1 & 0 & 0 & 0\\ 3 & 0 & 3 & 2 & 3 & 1 & 0 & 0\\ 3 & 0 & 0 & 3 & 2 & 3 & 1 & 0\\ 3 & 0 & 0 & 0 & 3 & 2 & 3 & 1 \end{pmatrix}\,. \tag*{(1)}\end{align}
Parent
Cousins
- Cyclic code — The octacode is a cyclic code over \(\mathbb{Z}_4\) with generator polynomial \(x^2+3x^2+2x+3\) extended by a parity check [4].
- \([7,4,3]\) Hamming code — The octacode reduces modulo-two to the \([8,4,4]\) extended Hamming code [4].
- Dual additive code — The octacode is self-dual with respect to the Euclidean inner product.
- Niemeier lattice code — The octacode can be used to construct a Niemeier lattice code via Construction \(A_4\) [5].
- \(E_8\) Gosset lattice code — The octacode yields the \(E_8\) Gosset lattice code via Construction \(A_4\) [6][7].
References
- [1]
- J. H. Conway and N. J. A. Sloane, “Self-dual codes over the integers modulo 4”, Journal of Combinatorial Theory, Series A 62, 30 (1993) DOI
- [2]
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
- [3]
- E. M. Rains and N. J. A. Sloane, “Self-Dual Codes”, (2002) arXiv:math/0208001
- [4]
- Self-Dual Codes and Invariant Theory (Springer-Verlag, 2006) DOI
- [5]
- A. Bonnecaze et al., “Niemeier lattices and Type II codes over Z4”, Discrete Mathematics 205, 1 (1999) DOI
- [6]
- A. Bonnecaze and P. Solé, “Quaternary constructions of formally self-dual binary codes and unimodular lattices”, Algebraic Coding 194 (1994) DOI
- [7]
- A. Bonnecaze, P. Sole, and A. R. Calderbank, “Quaternary quadratic residue codes and unimodular lattices”, IEEE Transactions on Information Theory 41, 366 (1995) DOI
Page edit log
- Victor V. Albert (2022-08-11) — most recent
Cite as:
“Octacode”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/octacode
Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/classical/rings/octacode.yml.