The decoding problem for block codes on the two-user binary adder channel is discussed, and a decoding algorithm based on a trellis diagram is introduced for the case where at least one of the component codes is a constant-weight code whose codewords belong to a known linear code. Both the noiseless and the noisy cases are considered. In the noisy case, it is possible to perform soft-decision decoding with a performance whose closeness to maximum likelihood decoding depends on the choice of the code pair. The resulting decoder complexity is proportional to the sum (rather than the product) of the number of codewords in the two component codes.