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.