Coding for the two-user binary adder channel (2-BAC) is investigated for the situation where one of the two codes is linear and the channel is noiseless. A simpler derivation of the transmission rate of linear strongly orthogonal codes is presented. Linear balanced strongly orthogonal codes are introduced, as well as repeated parity codes, together with their respective transmission rates. The Cabral algorithm for the linear construction of codes for the 2-BAC was implemented for the first time and a few of the codes obtained are presented in this paper.