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Coding schemes for the discrete memoryless T-user adder channel are investigated in this paper. First, the capacity region of the noiseless T-user adder channel, and the maximal achievable rate for T-user uniquely decodable codes are derived. Second, code constructions are presented, including a class of T-user uniquely decodable codes with rates asymptotically equal to the maximal achievable value. A decoding algorithm for these codes is also presented. Finally, a class of T-user error-correcting codes for the noisy adder channel is constructed.