Hexacode[1]

Description

The \([6,3,4]_{GF(4)}\) self-dual MDS code with generator matrix \begin{align} \begin{pmatrix} 1 & 0 & 0 & 1 & 1 & \omega\\ 0 & 1 & 0 & 1 & \omega & 1\\ 0 & 0 & 1 & \omega & 1 & 1 \end{pmatrix}~, \end{align} where \(GF(4) = \{0,1,\omega, \bar{\omega}\}\). Has connections to projective geometry, lattices [1] and conformal field theory [2].

Decoding

Bounded-distance decoder requiring at most 34 real operations [3].

Notes

See corresponding MinT database entry [4].

Parents

Cousins

  • Five-qubit perfect code — Applying the stabilizer-over-\(GF(4)\) construction to the hexacode yields a \([[6,0,4]]\) quantum code [8] corresponding to the six-qubit perfect state. The five-qubit code can be obtained from this code by tracing out a qubit [9].
  • Dual linear code — The hexacode is Euclidean and Hermitian self-dual.
  • Golay code — Extended Golay codewords can be obtained from hexacodewords [1]. The hexacode can be used to decode the extended Golay code [10]. There is also a connection between automoprhisms of the even Golay code and the holomorph of the hexacode [2].

References

[1]
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999). DOI
[2]
J. A. Harvey and G. W. Moore, “Moonshine, superconformal symmetry, and quantum error correction”, Journal of High Energy Physics 2020, (2020). DOI; 2003.13700
[3]
A. Vardy, “Even more efficient bounded-distance decoding of the hexacode, the Golay code, and the Leech lattice”, IEEE Transactions on Information Theory 41, 1495 (1995). DOI
[4]
Rudolf Schürer and Wolfgang Ch. Schmid. “Hexacode.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. http://mint.sbg.ac.at/desc_CHexa.html
[5]
T. Høholdt, J.H. Van Lint, and R. Pellikaan, 1998. Algebraic geometry codes. Handbook of coding theory, 1 (Part 1), pp.871-961.
[6]
J. Bierbrauer, Introduction to Coding Theory (Chapman and Hall/CRC, 2016). DOI
[7]
Self-dual Codes and Invariant Theory (Springer-Verlag, 2006). DOI
[8]
A. J. Scott, “Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions”, Physical Review A 69, (2004). DOI; quant-ph/0310137
[9]
F. Pastawski et al., “Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence”, Journal of High Energy Physics 2015, (2015). DOI; 1503.06237
[10]
V. Pless, “Decoding the Golay codes”, IEEE Transactions on Information Theory 32, 561 (1986). DOI

Zoo code information

Internal code ID: hexacode

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Zoo Code ID: hexacode

Cite as:
“Hexacode”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/hexacode
BibTeX:
@incollection{eczoo_hexacode, title={Hexacode}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/hexacode} }
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Permanent link:
https://errorcorrectionzoo.org/c/hexacode

Cite as:

“Hexacode”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/hexacode

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/classical/q-ary_digits/small/hexacode.yml.