The zero-error capacity region of the 2-user synchronous BAC is strictly smaller than its Shannon capacity region

New upper and lower bounds on the zero-error capacity region of the synchronous 2-user binary adder channel are given. The lower bound is in the form of a new uniquely decodable code with a sum rate higher than any previously reported such code. To derive the upper bound, we combine Van Tilborg's (1978) bound with Harper's (1966) isoperimetric inequality and a general information theoretic inequality. From this bound we conclude that the zero-error capacity region of the synchronous 2-user binary adder channel is strictly smaller than its associated Shannon capacity region.