Extended Golay Code

The extended Golay codes [1] are self-dual codes with parameters [24, 12, 8] over 2 and [12, 6, 6] over 3. They can be derived from extending a [23, 12, 7]-code over 2 and a [11, 6, 5]-code over 3, respectively, which are both cyclic, quadratic residue codes.

A generator for the extended Golay Code over 2 is

⟨10000000000⟩ 0 ⟨10100011101⟩ 1
00000000000 1 11111111111  0
,

a generator matrix for the extended Golay Code over 3 is

⟨10000⟩ 0 ⟨01221⟩ 1
00000 1 11111  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

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. “Extended Golay Code.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2008-04-04. http://mint.sbg.ac.at/desc_CGolay-extended.html

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