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Octacode[13]

Description

The unique self-dual linear \((8,4^4,6)_{\mathbb{Z}_4}\) code of Euclidean distance 8. Its shortened version is called the \((7,4^3,6)_{\mathbb{Z}_4}\) heptacode.

A generator matrix for this code is \begin{align} \left( \begin{array}{cccccccc} 1 & 1 & 2 & 1 & 3 & 0 & 0 & 0 \\ 1 & 0 & 1 & 2 & 1 & 3 & 0 & 0 \\ 1 & 0 & 0 & 1 & 2 & 1 & 3 & 0 \\ 1 & 0 & 0 & 0 & 1 & 2 & 1 & 3 \\ \end{array} \right)\,, \tag*{(1)}\end{align} and a generator matrix for the heptacode is \begin{align} \left( \begin{array}{ccccccc} 1 & 1 & 2 & 1 & 3 & 0 & 0 \\ 1 & 0 & 1 & 2 & 1 & 3 & 0 \\ 1 & 0 & 0 & 1 & 2 & 1 & 3 \\ \end{array} \right)\,. \tag*{(2)}\end{align}

Cousins

Primary Hierarchy

Parents
The octacode is a cyclic code over \(\mathbb{Z}_4\) with generator polynomial \(x^3+3x^2+2x+3\) extended by a parity check [5].
The octacode is self-dual over \(\mathbb{Z}_4\).
Octacode

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]
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
[5]
Self-Dual Codes and Invariant Theory (Springer-Verlag, 2006) DOI
[6]
A. Bonnecaze and P. Solé, “Quaternary constructions of formally self-dual binary codes and unimodular lattices”, Lecture Notes in Computer Science 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
[8]
W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, 2003) DOI
[9]
M. Shi, T. Honold, P. Sole, Y. Qiu, R. Wu, and Z. Sepasdar, “The Geometry of Two-Weight Codes Over ℤ\({}_{\text{\textit{p}}}\)\({}^{\text{\textit{m}}}\)”, IEEE Transactions on Information Theory 67, 7769 (2021) DOI
[10]
T. Honold and I. Landjev, “Linear Codes over Finite Chain Rings and Projective Hjelmslev Geometries”, Codes Over Rings 60 (2009) DOI
[11]
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
[12]
A. R. Hammons Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Solé, “The Z_4-Linearity of Kerdock, Preparata, Goethals and Related Codes”, (2002) arXiv:math/0207208
[13]
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..
[14]
Z. X. Wan, Quaternary Codes (WORLD SCIENTIFIC, 1997) DOI
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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} }
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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/edit/main/codes/classical/rings/over_zq/over_z4/linear_over_z4/self_dual/octacode.yml.