I need the optimal encoding protocol for 3 $\rightarrow$ 1 Classical RAC such that the receiver is able to retrieve any one of the initial bits, as well as the XOR combinations of those bits. ( If a, b, c are the bits in the string, then Bob would be asked to predict one of the seven possible functions, namely, a, b, c, a$\oplus$b, b$\oplus$c, a$\oplus$c, a$\oplus$b$\oplus$c).
The maximum that I got was 0.66 by encoding 000, 001, 110, 010, 101 as 0, and the rest of the strings as 1. Some other encodings have also given the same success probabilities.
But is this optimal? If so, how do I prove it?