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 a 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.Cousins
- Combinatorial design— Goethals codes form an infinite family of nonlinear binary codes supporting 3-designs [3; Table 5.1].
- Delsarte-Goethals (DG) code— Goethals 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][2; pg. 476]. However, the two codes are images of a pair of mutually dual linear codes over \(\mathbb{Z}_4\) under the Gray map [5].
- \(((2^m,2^{2^m−5m+1},8))\) Goethals-Preparata code— The \(((2^m,2^{2^m−5m+1},8))\) Goethals-Preparata code is constructed using the classical Goethals and Preparata codes [6,7]. 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 [8].
Member of code lists
Primary Hierarchy
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]
- V. D. Tonchev, “Codes and designs”, Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021), pp. 97-110 DOI
- [4]
- F. B. Hergert, “On the delsarte-goethals codes and their formal duals”, Discrete Mathematics 83, 249 (1990) DOI
- [5]
- A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Solé, “The Z_4-Linearity of Kerdock, Preparata, Goethals and Related Codes”, (2002) arXiv:math/0207208
- [6]
- M. Grassl and M. Rotteler, “Non-additive quantum codes from Goethals and Preparata codes”, 2008 IEEE Information Theory Workshop 396 (2008) arXiv:0801.2144 DOI
- [7]
- M. Grassl and M. Rotteler, “Quantum Goethals-Preparata codes”, 2008 IEEE International Symposium on Information Theory (2008) arXiv:0801.2150 DOI
- [8]
- S. Ling and P. Sole, “Nonadditive quantum codes from Z4-codes”, 2008 URL
- [9]
- P. Delsarte and J. M. Goethals, “Alternating bilinear forms over GF(q)”, Journal of Combinatorial Theory, Series A 19, 26 (1975) DOI
Page edit log
- Victor V. Albert (2026-06-08) — most recent
- Victor V. Albert (2024-01-11)
- Madhura Pankaja (2024-01-11)
Cite as:
“Goethals code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/goethals, arXiv:2606.11484