Description
The unique self-dual linear code of length 8 and Lee distance 6 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}
Parents
- Quaternary linear code over \(\mathbb{Z}_4\)
- Cyclic code — The octacode is a cyclic code over \(\mathbb{Z}_4\) with generator polynomial \(x^3+3x^2+2x+3\) extended by a parity check [4].
- Self-dual code over \(R\) — The octacode is self-dual over \(\mathbb{Z}_4\).
Cousins
- \([7,4,3]\) Hamming code — The octacode can be obtained by Hensel-lifting the \([7,4,3]\) Hamming code to \(\mathbb{Z}_4\) [5].
- \([8,4,4]\) extended Hamming code — The octacode reduces modulo-two to the \([8,4,4]\) extended Hamming code [4].
- \(\Lambda_{24}\) Leech lattice — The Leech lattice can be constructed via the Turyn construction and the holy construction using the octacode as the glue code [6].
- Niemeier lattice — The octacode can be used to construct a Niemeier lattice code via Construction \(A_4\) [7].
- \(E_8\) Gosset lattice — The octacode yields the \(E_8\) Gosset lattice code via Construction \(A_4\) [8,9].
- Nordstrom-Robinson (NR) code — The NR code is the image of the octacode under the Gray map [10].
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. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Sole, “The Z/sub 4/-linearity of Kerdock, Preparata, Goethals, and related codes”, IEEE Transactions on Information Theory 40, 301 (1994) DOI
- [6]
- Feng-Wen Sun and H. C. A. van Tilborg, “The Leech lattice, the octacode, and decoding algorithms”, IEEE Transactions on Information Theory 41, 1097 (1995) DOI
- [7]
- A. Bonnecaze, P. Gaborit, M. Harada, M. Kitazume, and P. Solé, “Niemeier lattices and Type II codes over Z4”, Discrete Mathematics 205, 1 (1999) DOI
- [8]
- A. Bonnecaze and P. Solé, “Quaternary constructions of formally self-dual binary codes and unimodular lattices”, Lecture Notes in Computer Science 194 (1994) DOI
- [9]
- A. Bonnecaze, P. Sole, and A. R. Calderbank, “Quaternary quadratic residue codes and unimodular lattices”, IEEE Transactions on Information Theory 41, 366 (1995) DOI
- [10]
- Forney Jr GD, Sloane NJ, Trott MD. The Nordstrom-Robinson code is the binary image of the octacode. In Coding and Quantization: DIMACS/IEEE workshop 1992 Oct 19 (pp. 19-26). Amer. Math. Soc..
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