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
- \([7,4,3]\) Hamming code— The octacode can be obtained by Hensel-lifting the \([7,4,3]\) Hamming code to \(\mathbb{Z}_4\) [4].
- \([8,4,4]\) extended Hamming code— The mod-two reduction of the octacode is the \([8,4,4]\) extended Hamming code [5].
- \(E_8\) Gosset lattice— The octacode yields the \(E_8\) Gosset lattice via Construction \(A_4\) [6,7][8; Exam. 12.5.13].
- Projective geometry code— Columns of the heptacode's (octacode's) generator matrix represent the seven (eight) points of a hyperoval (8-arc) in the projective Hjelmslev plane \(PHG(2,\mathbb{Z}_4)\) (\(PHG(3,\mathbb{Z}_4)\)) [10][9; Exam. 5].
- \([7,3,4]\) simplex code— Codewords of the heptacode with entries 0 and 2 are of the form \(2c\), where \(c\) is a codeword of the \([7,3,4]\) simplex code [9; Exam. 5].
- \(\Lambda_{24}\) Leech lattice— The Leech lattice can be constructed via the Turyn construction and the holy construction using the octacode as the glue code; one of these constructions uses eight copies of the \(D_3\) fcc lattice [11,12].
- Nordstrom-Robinson (NR) code— The NR code is the image of the octacode under the Gray map [13,14]. The \((14, 64, 6)\) shortened NR code is the image of the heptacode under the Gray map [9; Exam. 5].
Member of code lists
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
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- [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
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- Self-Dual Codes and Invariant Theory (Springer-Verlag, 2006) DOI
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- A. Bonnecaze and P. Solé, “Quaternary constructions of formally self-dual binary codes and unimodular lattices”, Lecture Notes in Computer Science 194 (1994) DOI
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- A. Bonnecaze, P. Sole, and A. R. Calderbank, “Quaternary quadratic residue codes and unimodular lattices”, IEEE Transactions on Information Theory 41, 366 (1995) DOI
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- 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
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