Hexacode[1][2]

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}~, \tag*{(1)}\end{align} where \(GF(4) = \{0,1,\omega, \bar{\omega}\}\). Has connections to projective geometry, lattices [2] and conformal field theory [3].

Decoding

Bounded-distance decoder requiring at most 34 real operations [4].

Notes

See corresponding MinT database entry [5].

Parents

  • Hyperoval code — Columns of hexacode's generator matrix represent the six homogeneous coordinates of a hyperoval in the projective plane \(PG(2,4)\) ([6], pg. 289).
  • Evaluation AG code — The hexacode is an evaluation AG code over \(GF(4) = \{0,1,\omega, \bar{\omega}\}\) with \(\cal X\) defined by \(x^2 y + \omega y^2 z + \bar{\omega} z^2 x = 0\) ([7], Ex. 2.77).
  • \(q\)-ary quadratic-residue (QR) code — The hexacode is the smallest example of an extended quadratic residue code of Type \(4^H\) ([8], Sec. 2.4.6).
  • Denniston code — A version of the hexacode is recovered for Dennison code parameters \(i=1\) and \(a=2\) [6].
  • Maximum distance separable (MDS) code

Cousins

  • Five-qubit perfect code — Applying the stabilizer-over-\(GF(4)\) construction to the hexacode yields a \([[6,0,4]]\) quantum code [9] corresponding to the six-qubit perfect state. The five-qubit code can be obtained from this code by tracing out a qubit [10].
  • Dual linear code — The hexacode is Euclidean and Hermitian self-dual.
  • Golay code — Extended Golay codewords can be obtained from hexacodewords [2]. The hexacode can be used to decode the extended Golay code [11]. There is also a connection between automoprhisms of the even Golay code and the holomorph of the hexacode [3].
  • Coxeter-Todd \(K_{12}\) lattice code — The hexacode can be used to obtain the Coxeter-Todd \(K_{12}\) lattice code [12; Ex. 10.5.6].

References

[1]
K. A. Bush, “Orthogonal Arrays of Index Unity”, The Annals of Mathematical Statistics 23, 426 (1952) DOI
[2]
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
[3]
J. A. Harvey and G. W. Moore, “Moonshine, superconformal symmetry, and quantum error correction”, Journal of High Energy Physics 2020, (2020) arXiv:2003.13700 DOI
[4]
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
[5]
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
[6]
J. Bierbrauer, Introduction to Coding Theory (Chapman and Hall/CRC, 2016) DOI
[7]
T. Høholdt, J.H. Van Lint, and R. Pellikaan, 1998. Algebraic geometry codes. Handbook of coding theory, 1 (Part 1), pp.871-961.
[8]
Self-Dual Codes and Invariant Theory (Springer-Verlag, 2006) DOI
[9]
A. J. Scott, “Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions”, Physical Review A 69, (2004) arXiv:quant-ph/0310137 DOI
[10]
F. Pastawski et al., “Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence”, Journal of High Energy Physics 2015, (2015) arXiv:1503.06237 DOI
[11]
V. Pless, “Decoding the Golay codes”, IEEE Transactions on Information Theory 32, 561 (1986) DOI
[12]
T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
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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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“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/easy/hexacode.yml.