Show that every word of weight $4$ in $\mathbb{F}_{2}^{23}$ is of distance $3$ from exactly one codeword in the binary Golay code $G_{23}$.

Question

Show that every word of weight $4$ in $\mathbb{F}_{2}^{23}$ is of distance $3$ from exactly one codeword in the binary Golay code $G_{23}$.

I can only show that the word is necessarily distance 3 from any codeword but do not know how to show the existence of such a word, and why it is the only one. Any help will be greatly appreciated! Thank you!

Answer 1

Since the binary Golay code is a perfect code of distance 7, a word $\textbf{x}$ of weight 4 must be contained in some sphere of radius 3 with the center being a codeword $\textbf{c}$. Since the weight of $\textbf{c}$ is at least 7. we conclude that the weight of $\textbf{c}$ is exactly 7.

Because there is no intersection between any two spheres, $\textbf{c}$ is the unique codeword of weight 7 of distance 3 from $\textbf{x}$.