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  "body": "\n## Description\n\nThe extended binary Golay code $G_{24}$ is a linear code of length $24$ over the field $\\F_2$, of dimension $12$ and minimum distance $8$.\nSo it has $2^{12} = 4096$ codewords, and any two differ in at least $8$ of the $24$ positions; as an [[error-correcting code|X122TKFP9G]] it corrects $3$ errors.\nDeleting one coordinate gives the perfect binary Golay code $G_{23}$, of length $23$, dimension $12$ and minimum distance $7$.\nCheck that $G_{23}$ is perfect: a Hamming ball of radius $3$ in $\\F_2^{23}$ contains $1 + 23 + 253 + 1771 = 2048 = 2^{11}$ words.\nThe $2^{12}$ balls around the codewords are disjoint, since $7 \\ge 2 \\cdot 3 + 1$, and together they contain $2^{12} \\cdot 2^{11} = 2^{23}$ words, which is all of $\\F_2^{23}$.\n\n## Remarks\n\nThe weights of the codewords of $G_{24}$ are $0, 8, 12, 16, 24$, with $1, 759, 2576, 759, 1$ codewords of each weight.\nThe $759$ codewords of weight $8$, seen as $8$-element subsets of the $24$ positions, are the blocks of the [[Steiner system|7BPSC4T4M3]] $S(5, 8, 24)$.\nThe automorphism group of $G_{24}$ is the Mathieu group $M_{24}$, one of the sporadic simple groups.\nBesides the binary codes there is also a ternary Golay code.\n\n## Explanations\n\n### L1 human\n\nThe binary Golay code is a set of $4096$ strings of $24$ bits, chosen so that any two differ in at least $8$ bits.\nIf a string is damaged in up to $3$ bits, the original string can still be recovered.\nIts shorter version of length $23$ is perfect: the spheres of radius $3$ around its codewords fill the whole space with no gap and no overlap.\n",
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