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Golay Code

The Golay code is a perfect linear error-correcting code. There are two essentially distinct versions of the Golay code: a binary version and a ternary version.

The binary version is a binary linear code consisting of codewords of length 23 and minimum distance 7. The ternary version is a ternary linear code, consisting of codewords of length 11 with minimum distance 5.

A parity check matrix for the binary Golay code is given by the matrix , where is the identity matrix and is the matrix

By adding a parity check bit to each codeword in , the extended Golay code , which is a nearly perfect binary linear code, is obtained. The automorphism group of is the Mathieu group .

A second generator is the adjacency matrix for the icosahedron, with appended, where is a unit matrix and is an identity matrix.

A third generator begins a list with the 24-bit 0 word (000...000) and repeatedly appends first 24-bit word that has eight or more differences from all words in the list. Conway and Sloane list many further methods.

Amazingly, Golay's original paper was barely a half-page long but has proven to have deep connections to group theory, graph theory, number theory, combinatorics, game theory, multidimensional geometry, and even particle physics.

Code, Coding Theory, Error-Correcting Code, Linear Code, Mathieu Groups, Nearly Perfect Code, Parity Check Matrix, Perfect Code

Portions of this entry contributed by David Terr

This entry contributed by Ed Pegg, Jr. (author's link)

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References

Conway, J. H. and Sloane, N. J. A. Sphere Packings, Lattices, and Groups, 3rd ed. New York: Springer, 1999.Golay, M. J. E. "Notes on Digital Coding." Proc. IRE 37, 657, 1949.Heumann, S. "Golay Codes." http://www.mdstud.chalmers.se/~md7sharo/coding/main/node34.html.van Lint, J. H. An Introduction to Coding Theory, 2nd ed. New York: Springer-Verlag, 1992.

Golay Code

Cite this as:

Pegg, Ed Jr.; Terr, David; and Weisstein, Eric W. "Golay Code." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GolayCode.html