Introduction to Cryptography with Coding Theory, 3rd Edition

24.6 Golay Codes

Two of the most famous binary codes are the Golay codes $G_{23}$ and $G_{24}$. The [24, 12, 8] extended Golay code $G_{24}$ was used by the Voyager I and Voyager II spacecrafts during 1979–1981 to provide error correction for transmission back to Earth of color pictures of Jupiter and Saturn. The (nonextended) Golay code $G_{23}$, which is a [23, 12, 7] code, is closely related to $G_{24}$. We shall construct $G_{24}$ first, then modify it to obtain $G_{23}$. There are many other ways to construct the Golay codes. See [MacWilliams-Sloane].

The generating matrix for $G_{24}$ is the $12\times 24$ matrix $G=$

$$\left(\begin{array}{cccccccccccccccccccccccc}1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 1& 1& 0& 1& 1& 1& 0& 0& 0& 1& 0\\ 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 1& 1& 0& 1& 1& 1& 0& 0& 0& 1\\ 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 1& 0& 1& 1& 0& 1& 1& 1& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 1& 0& 1& 1& 0& 1& 1& 1& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 1& 0& 1& 1& 0& 1& 1& 1& 0\\ 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 1& 0& 1& 1& 0& 1& 1& 1\\ 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 1& 1& 0& 0& 0& 1& 0& 1& 1& 0& 1& 1\\ 0\end{array}\right)$$