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 4-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 coding 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 = F²⁴
₂ 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 F²³
₂.

The automorphism group of the perfect binary Golay code G₂₃ (meaning the subgroup of the group S₂₃ of permutations of the coordinates of F²³
₂ which leave G₂₃ invariant), is the Mathieu group ${\displaystyle M_{23}}$. The automorphism group of the extended binary Golay code is the Mathieu group ${\displaystyle M_{24}}$, of order 2¹⁰ × 3³ × 5 × 7 × 11 × 23. ${\displaystyle M_{24}}$ is transitive on octads and on dodecads. The other Mathieu groups occur as stabilizers of one or several elements of W.

There is a single word of weight 24, which is a 1-dimensional invariant subspace. ${\displaystyle M_{24}}$ therefore has an 11-dimensional irreducible representation on the field with 2 elements. In addition, since the binary golay code is a 12-dimensional subspace of a 24-dimensional space, ${\displaystyle M_{24}}$ also acts on the 12-dimensional quotient space, called the binary Golay cocode. A word in the cocode is in the same coset as a word of length 0, 1, 2, 3, or 4. In the last case, 6 (disjoint) cocode words all lie in the same coset. There is an 11-dimensional invariant subspace, consisting of cocode words with odd weight, which gives ${\displaystyle M_{24}}$ a second 11-dimensional representation on the field with 2 elements.

It is convenient to use the "Miracle Octad Generator" format, with coordinates in an array of 4 rows, 6 columns. Addition is taking the symmetric difference. All 6 columns have the same parity, which equals that of the top row.

A partition of the 6 columns into 3 pairs of adjacent ones constitutes a trio. This is a partition into 3 octad sets. A subgroup, the projective special linear group PSL(2,7) x S₃ of a trio subgroup of M₂₄ is useful for generating a basis. PSL(2,7) permutes the octads internally, in parallel. S₃ permutes the 3 octads bodily.