Goethals code[1] 

Description

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].

Rate

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

Parent

Cousin

  • 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].

References

[1]
J. M. Goethals, “Two dual families of nonlinear binary codes”, Electronics Letters 10, 471 (1974) DOI
[2]
F. J. MacWilliams and N. J. A. Sloane. The theory of error correcting codes. Elsevier, 1977.
[3]
P. Delsarte and J. M. Goethals, “Alternating bilinear forms over GF(q)”, Journal of Combinatorial Theory, Series A 19, 26 (1975) DOI
[4]
F. B. Hergert, “On the delsarte-goethals codes and their formal duals”, Discrete Mathematics 83, 249 (1990) DOI
[5]
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
[6]
A. R. Hammons Jr. et al., “The Z_4-Linearity of Kerdock, Preparata, Goethals and Related Codes”, (2002) arXiv:math/0207208
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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
BibTeX:
@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.