 Solution

A compact code or optimal prefix code minimizes the expected codeword length ${\sum }_x{p}_x{\ell }_x$ for the given source probabilities and coding alphabet. The relevant comparison here is among binary prefix codes. The minimum is also the minimum among uniquely decodable codes, since the Kraft-McMillan inequality permits every uniquely decodable length list to be realized by a prefix code with the same lengths.