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.
Page edit log
- Victor V. Albert (2022-08-09) — most recent
Cite as:
“Hexacode”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/hexacode