Golay code

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.^[ref]

The binary length-twenty-three code corrects three errors. A radius-three Hamming ball contains $1+23+\left(\genfrac{}{}{0ex}{}{23}{2}\right)+\left(\genfrac{}{}{0ex}{}{23}{3}\right)=2048=2^{11}$ 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.