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].


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


See corresponding MinT database entry [5].


  • 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


  • 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].


K. A. Bush, “Orthogonal Arrays of Index Unity”, The Annals of Mathematical Statistics 23, 426 (1952) DOI
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) arXiv:2003.13700 DOI
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
J. Bierbrauer, Introduction to Coding Theory (Chapman and Hall/CRC, 2016) DOI
T. Høholdt, J.H. Van Lint, and R. Pellikaan, 1998. Algebraic geometry codes. Handbook of coding theory, 1 (Part 1), pp.871-961.
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) arXiv:quant-ph/0310137 DOI
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
V. Pless, “Decoding the Golay codes”, IEEE Transactions on Information Theory 32, 561 (1986) DOI
T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
Page edit log

Your contribution is welcome!

on github.com (edit & pull request)

edit on this site

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} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:

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/easy/hexacode.yml.