Octacode[1][2][3]

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

Your contribution is welcome!

on github.com (edit & pull request)

edit on this site

Zoo Code ID: octacode

Cite as:
“Octacode”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/octacode
BibTeX:
@incollection{eczoo_octacode, title={Octacode}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/octacode} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:
https://errorcorrectionzoo.org/c/octacode

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.