I rephrased the question slightly for clarity after you posted this: You're looking for the largest value N such that you can guarantee that you get at least N questions correct. Since log(5400)/log(2) is only ~12.3987, your approach only counts as 12 correct answers.

(You can do better.)

We can start with the prior that there will probably be a similar number of trues and falses. We are mapping 23 bits of state into 12 bits, we can leave out those states where T>>F or vice versa. If the test turns out to be one of those unexpected states, pick the answer that gives the best score.

Edit: Rereading, it looks like we're optimizing the worst case, rather than average. So we're looking for a 12:23 map where at most n bits are inaccurate, minimizing n. I'm sure there's a signals alg that does this...