The Golay binary code is one of two linear error codes linked together:
- perfect binary Golei code - perfect binary code with parameters , or
- extended Golay binary code obtained from the perfect by adding the parity bit and having parameters .
![{\ displaystyle [23,12,7]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc79baa92efb12afed9ec31c4bf8dd7a7669a3f4)
![[24,12,8]](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a5346c89f3c9e9c7beba1655a3fba8122150193)
| Perfect Golei Binary Code | |
|---|---|
| Named after | |
| Type of | linear block code |
| Block length | 23 |
| Message Length | 12 |
| Share | 12/23 ~ 0.522 |
| Distance | 7 |
| Alphabet size | 2 |
| Designation | |
![{\ displaystyle [23,12,7] _ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c17f5f280c4e4796b177454dd9e84c3b7ade8325)
| Expanded Boley Code | |
|---|---|
| Named after | |
| Type of | linear block code |
| Block length | 24 |
| Message Length | 12 |
| Share | 12.24 = 0.5 |
| Distance | eight |
| Alphabet size | 2 |
| Designation | |
![{\ displaystyle [24,12,8] _ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94468240ab43ea76dd2a763b3ae1442fdc6bdded)
Properties
- The perfect Golei code corrects 3 errors.
- The extended Golay code is twice even (the norm of any vector is divided by 4), and is unimodular (the dimension is half the dimension of space).
- The minimum norm of a nonzero vector of the extended Golay code is 8. The dimension 24 is the first in which a twice-even unimodular code may not have a norm vector 4.
- Automated Group of Extended Golei Code - Mathieu Group .
- Sets of units of the norm 8 vectors of the extended Golei code form the Steiner system .
Application
The Golei code was used during the Voyager program when transmitting color images of Jupiter and Saturn by the Voyager-1 and Voyager-2 devices.
See also
- Lich Lattice
Notes
Links
- Pegg, Ed Jr .; Terr, David; and Weisstein, Eric W. Golay Code on Wolfram MathWorld .
- Golay, Marcel JE Notes on Digital Coding (English) // Proc. IRE : journal. - 1949. - Vol. 37 . - P. 657 .
- Curtis, RT A new combinatorial approach to M 24 // Math. Proc. Camb. Phil. Soc. : journal. - 1976. - Vol. 79 . - P. 25β42 . - DOI : 10.1017 / S0305004100052075 .
- Griess, Robert L. Twelve Sporadic Groups. - Springer, 1998 .-- P. 167. - ISBN 9783540627784 .
- Thompson, Thomas M. From Error Correcting Codes through Sphere Packings to Simple Groups. - Mathematical Association of America , 1983. - Vol. 21. - ISBN 9780883850237 .