The \([6,3,4]_4\) 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].

Puncturing the code yields the perfect \([5,3,3]_4\) quaternary Hamming code known as the shortened hexacode or shorter hexacode [4]. Both codes are sometimes refereed to as Golay codes over \(GF(4)\).


Bounded-distance decoder requiring at most 34 real operations [5].


See [6; Sec. 10.3] for an exposition.See corresponding MinT database entry [7].




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Zoo Code ID: hexacode

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

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