I would say that the lower bound is 18. Because we want to represent a 23 bits string by 12 bits codewords, the max distance between 2 codewords is 11 bits. S

I would say that the lower bound is 18. Because we want to represent a 23 bits string by 12 bits codewords, the max distance between 2 codewords is 11 bits. So the protocol would be to agree on a dictionary of 12 bits codewords and send the closest codeword to the actual answer. The max hamming distance between any answers and a codeword would be floor(11/2) = 5. Which corresponds to 18 correctly answered questions.