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
- Hergert code — Goethals codes are equivalent to Hergert codes for \(r=3\) [3; Thm. 2].
Cousins
- 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].
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
- [7]
- M. Grassl and M. Rotteler, “Non-additive quantum codes from Goethals and Preparata codes”, 2008 IEEE Information Theory Workshop (2008) arXiv:0801.2144 DOI
- [8]
- M. Grassl and M. Rotteler, “Quantum Goethals-Preparata codes”, 2008 IEEE International Symposium on Information Theory (2008) arXiv:0801.2150 DOI
- [9]
- S. Ling and P. Sole. 2008. Nonadditive quantum codes from Z4-codes. http://hal.archives-ouvertes.fr/hal-00338309/fr/.
Page edit log
- Victor V. Albert (2024-01-11) — most recent
- Madhura Pankaja (2024-01-11)
Cite as:
“Goethals code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/goethals