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].


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


See corresponding MinT database entry [4].



  • 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].


J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999). DOI
J. A. Harvey and G. W. Moore, “Moonshine, superconformal symmetry, and quantum error correction”, Journal of High Energy Physics 2020, (2020). DOI; 2003.13700
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
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
T. Høholdt, J.H. Van Lint, and R. Pellikaan, 1998. Algebraic geometry codes. Handbook of coding theory, 1 (Part 1), pp.871-961.
J. Bierbrauer, Introduction to Coding Theory (Chapman and Hall/CRC, 2016). DOI
Self-dual Codes and Invariant Theory (Springer-Verlag, 2006). DOI
A. J. Scott, “Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions”, Physical Review A 69, (2004). DOI; quant-ph/0310137
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
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
@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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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.