Golay Code
The extended Golay codes [1] are self-dual codes with parameters [24, 12, 8] over ℤ2 and [12, 6, 6] over ℤ3. They can be derived from extending a [23, 12, 7]-code over ℤ2 and a [11, 6, 5]-code over ℤ3, respectively, which are both cyclic, quadratic residue codes.
A generator for the extended Golay Code over ℤ2 is
| ⟨10000000000⟩ 0 ⟨10100011101⟩ 1 |
| 00000000000Â 1Â 11111111111Â 0 |
a generator matrix for the extended Golay Code over ℤ3 is
| ⟨10000⟩ 0 ⟨01221⟩ 1 |
| 00000Â 1Â 11111Â 0 |
Optimality
The Golay codes meet the Hamming bound with equality and are therefore perfect codes. The extended Golay codes are nearly perfect codes.
See also
Golay code at
[2, pp. 258]
References
| [1] | Marcel J. E. Golay. Notes on digital coding. Proceedings of the IEEE, 37:657, 1949. |
| [2] | Jürgen Bierbrauer. Introduction to Coding Theory. Discrete Mathematics and its Applications. Chapman & Hall/CRC, Boca Raton, London, New York, Washington D.C., 2004. MR2079734 (2005f:94001) |
Copyright
Copyright © 2004, 2005, 2006, 2007, 2008, 2009, 2010 by Rudolf Schürer and Wolfgang Ch. Schmid.
Cite this as: Rudolf Schürer and Wolfgang Ch. Schmid. “Golay Code.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2024-09-05.
http://mint.sbg.ac.at/desc_CGolay.html