Consider five distinct binary vectors $X_{1}, \ldots, X_{5}$ each of length $10$. Let
\[d_{i j}=\sum_{k=1}^{10}\left(X_{i k} \text { XOR } X_{j k}\right),\]
(i.e., $d_{i j}$ is the number of coordinates where $X_{i}$ and $X_{j}$ differ) be the Hamming distance between $X_{i}$ and $X_{j}$ and let $d=\min _{i, j=1, \ldots, 5, i \neq j} d_{i j}$. Which of the following is $\text{TRUE}?$ [Hint: Look at the first two entries of $X_{1}$ to $X_{5}$, and argue about the result noting that there are five binary vectors.]
- $d=10$
- $d=9$
- $d=8$
- $d<8$
- Information is not sufficient