Suppose that Alice’s messages are m_{1}, \ldots, m_{n}. Suppose that Alice picks a random binary string s of length n, and then constructs a packet m_{s} by \oplus ing every message for which s has a 1. For example, if n=4, m_{1010}=m_{1} \oplus m_{3}. Each turn, Alice will generate a random s and send Bob the packet m_{s}.
Let e_{i} be the binary string of length n with a 1 at the i the entry and 0s elsewhere. Show that, if she can write e_{i}=s_{1} \oplus \cdots \oplus s_{k}, then m_{i}=m_{s_{1}} \oplus \cdots \oplus m_{s_{k}}.