Golay code

Description

The extended binary Golay code $G_{24}$ is a linear code of length $24$ over the field $\F_2$, of dimension $12$ and minimum distance $8$. So it has $2^{12} = 4096$ codewords, and any two differ in at least $8$ of the $24$ positions; as an error-correcting code it corrects $3$ errors. Deleting one coordinate gives the perfect binary Golay code $G_{23}$, of length $23$, dimension $12$ and minimum distance $7$. Check that $G_{23}$ is perfect: a Hamming ball of radius $3$ in $\F_2^{23}$ contains $1 + 23 + 253 + 1771 = 2048 = 2^{11}$ words. The $2^{12}$ 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 $\F_2^{23}$.