  We present a general method to modify existing uniquely decodable codes in
the $T$-user binary adder channel. If at least one of the original constituent
codes does not have average weight exactly half of the dimension, then our
method produces a new set of constituent codes in a higher dimension, with a
strictly higher rate. Using our method we improve the highest known rate for
the $T$-user binary adder channel for all $T \geq 2$. This information theory
problem is equivalent to co-Sidon problems initiated by Lindstr{\"o}m in the
1960s, and also the multi-set union-free problem. Our results improve the known
lower bounds in these settings as well.
