A uniquely decodable code pair (C, S) is considered for the two-user binary adder channel. When the first code C is linear, a lower bound of |S| is formulated and a uniquely decodable code pair (C, S) is presented. When a rate R1 of C is less than 1/3, a rate R2of S is greater than the best rate known previously.

A uniquely decodable code (C1, C2, C3) is investigated for the three-user binary adder channel. The uniquely decodable code is constructed as follows: If C1 is an (n, k) linear code with a generator matrix, C2 is a coset of C1 and C3 is a set of all coset leaders, then the code (C1, C2, C3) is uniquely decodable and its total rate is equal to 1k/n, n2k. This code is easily decodable.

A uniquely decodable (UD) code pair (C, S) is considered for the two-user binary adder channel. For a class of linear codes C, the maximum independent set of the graph associated with C, which is the second code S, is evaluated. When the rate R1 of C is less than 0.5, there exist UD codes (C, S)'s such that the rate R2 of S exceeds the Khachatrian's and Guo's results in amount.