Goethals code[1] 


Member of a family of \((2^m,2^{2^m-3m+1},8)\) binary nonlinear codes for \(m \geq 6\) that generalizes the Preparata codes. The code can be constructed as disjoint union of cosets of a certain linear code [2; Ch. 15].


The rate is \({2^m -3m +1}/2^m\), going to 1 as block length goes to infinity.



  • Delsarte-Goethals (DG) code — Hergert codes for a given \(m\) are duals of DG\((m,1/2(m-2))\) codes in that their distance distribution is equal to the MacWilliams transform of the distance distribution of the DG codes [4]. However, the two codes are images of a pair of mutually dual linear codes over \(\mathbb{Z}_4\) under the Gray map [5,6].
  • \(((2^m,2^{2^m−5m+1},8))\) Goethals-Preparata code — The \(((2^m,2^{2^m−5m+1},8))\) Goethals-Preparata code is constructed using the classical Goethals and Preparata codes [7,8]. A construction using the \(\mathbb{Z}_4\) versions of the Goethals and Preparata codes and the Gray map yields qubit code families with similar parameters [9].


J. M. Goethals, “Two dual families of nonlinear binary codes”, Electronics Letters 10, 471 (1974) DOI
F. J. MacWilliams and N. J. A. Sloane. The theory of error correcting codes. Elsevier, 1977.
P. Delsarte and J. M. Goethals, “Alternating bilinear forms over GF(q)”, Journal of Combinatorial Theory, Series A 19, 26 (1975) DOI
F. B. Hergert, “On the delsarte-goethals codes and their formal duals”, Discrete Mathematics 83, 249 (1990) DOI
A. R. Hammons et al., “The Z/sub 4/-linearity of Kerdock, Preparata, Goethals, and related codes”, IEEE Transactions on Information Theory 40, 301 (1994) DOI
A. R. Hammons Jr. et al., “The Z_4-Linearity of Kerdock, Preparata, Goethals and Related Codes”, (2002) arXiv:math/0207208
M. Grassl and M. Rotteler, “Non-additive quantum codes from Goethals and Preparata codes”, 2008 IEEE Information Theory Workshop (2008) arXiv:0801.2144 DOI
M. Grassl and M. Rotteler, “Quantum Goethals-Preparata codes”, 2008 IEEE International Symposium on Information Theory (2008) arXiv:0801.2150 DOI
S. Ling and P. Sole. 2008. Nonadditive quantum codes from Z4-codes. http://hal.archives-ouvertes.fr/hal-00338309/fr/.
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Zoo Code ID: goethals

Cite as:
“Goethals code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/goethals
@incollection{eczoo_goethals, title={Goethals code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/goethals} }
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Cite as:

“Goethals code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/goethals

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/bits/nonlinear/gray_map/duals/goethals.yml.