Binary Golay code[1]
Description
A \([23, 12, 7]\) perfect binary linear code with connections to the theory of sporadic simple groups. Adding a parity bit to the code results in the \([24, 12, 8]\) extended Golay code. The codespace is a 12-dimensional linear subspace of \(GF(2)^{23}\), or \(GF(2)^{24}\) in the extended case.
The \([23, 12, 7]\) code can be constructed with the polynomial \(x^{11} + x^{10} + x^6 + x^5 + x^4 + x^2 + 1\) over \(GF(2)\). This representation of the Golay code is cyclic. Alternatively, one can use the Great Dodecahedron to generate codewords by placing message bits on the faces and calculating the parity bits that live on the 12 vertices of the inner icosahedron.
Protection
Rate
Decoding
Realizations
Notes
Parents
- Linear binary code
- Perfect code
- Quadratic-residue code — The collection of nonzero quadratic residues modulo 23 can be used to construct a generator polynomial of the Golay code.
Zoo code information
References
- [1]
- Golay, M. J. E. "Notes on Digital Coding." Proc. IRE 37, 657, 1949.
- [2]
- E. R. Berlekamp and L. Welch, Error Correction of Algebraic Block Codes. U.S. Patent, Number 4,633,470 1986.
- [3]
- A. J. VITERBI, “Error Bounds for Convolutional Codes and an Asymptotically Optimum Decoding Algorithm”, The Foundations of the Digital Wireless World 41 (2009). DOI
- [4]
- B. Honary and G. Markarian, “New simple encoder and trellis decoder for Golay codes”, Electronics Letters 29, 2170 (1993). DOI
- [5]
- E. C. Stone, “The Voyager 2 encounter with Uranus”, Journal of Geophysical Research: Space Physics 92, 14873 (1987). DOI
Cite as:
“Binary Golay code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/golay
Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/classical/bits/golay.yml.