= Golay code
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The name <Golay code> refers to closely related exceptional <linear codes>. The <binary Golay code> has parameters $[23,12,7]_2$: length twenty-three, <dimension> twelve over $\mathbb F_2$, and <minimum distance> seven. Adding an overall parity coordinate gives the <extended binary Golay code> $[24,12,8]_2$. There is also a ternary $[11,6,5]_3$ code, whose extension has parameters $[12,6,6]_3$; specifying the alphabet and extension avoids ambiguity.<Golay codes in the Error Correction Zoo>[Binary and ternary Golay code entries, parameters and perfect-code properties]{ref}
The binary length-twenty-three code corrects three errors. A radius-three <Hamming ball> contains $1+23+\binom{23}{2}+\binom{23}{3}=2048=2^{11}$ words. Its $2^{12}$ disjoint decoding balls therefore cover all $2^{23}$ received words exactly: this is a <perfect code>. The extended code has a larger <minimum distance>, but its radius-three balls do not fill the entire larger word space. These arithmetic properties explain why the codes are useful beyond their historical names.
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