Binary Golay code

Extended binary Golay code
Its generator matrix
Named after	Marcel J. E. Golay
Classification
Type	Linear block code
Block length	24
Message length	12
Rate	12/24 = 0.5
Distance	8
Alphabet size	2
Notation	${\displaystyle [24,12,8]_{2}}$-code

Perfect binary Golay code
Named after	Marcel J. E. Golay
Classification
Type	Linear block code
Block length	23
Message length	12
Rate	12/23 ~ 0.522
Distance	7
Alphabet size	2
Notation	${\displaystyle [23,12,7]_{2}}$-code

In mathematics and electronics engineering, a binary Golay code is a type of linear error-correcting code used in digital communications. The binary Golay code, along with the ternary Golay code, has a particularly deep and interesting connection to the theory of finite sporadic groups in mathematics. These codes are named in honor of Marcel J. E. Golay whose 1949 paper introducing them has been called, by E. R. Berlekamp, the "best single published page" in coding theory.

There are two closely related binary Golay codes. The extended binary Golay code, G₂₄ (sometimes just called the "Golay code" in finite group theory) encodes 12 bits of data in a 24-bit word in such a way that any 3-bit errors can be corrected or any 7-bit errors can be detected. The other, the perfect binary Golay code, G₂₃, has codewords of length 23 and is obtained from the extended binary Golay code by deleting one coordinate position (conversely, the extended binary Golay code is obtained from the perfect binary Golay code by adding a parity bit). In standard code notation the codes have parameters [24, 12, 8] and [23, 12, 7], corresponding to the length of the codewords, the dimension of the code, and the minimum Hamming distance between two codewords, respectively.

In mathematical terms, the extended binary Golay code G₂₄ consists of a 12-dimensional linear subspace W of the space V=F₂²⁴ of 24-bit words such that any two distinct elements of W differ in at least 8 coordinates. W is called a linear code because it is a vector space. In all, W comprises 4096 = 2¹² elements.

The binary Golay code, G₂₃ is a perfect code. That is, the spheres of radius three around code words form a partition of the vector space. G₂₃ is a 12-dimensional subspace of the space F₂²³.