Can Chinese remainder theorem be used to solve this problem in multiuser coding?

We have two transmitters sending integers $q,q'>0$ to a common receiver. The duty of the receiver is to recover both $q,q'$. Assume the integers below are sufficiently large.

The received integer is $uq+u'q'+n$ where every letter is an integer above $0$. Suppose we know $u,u'$ and the unknowns are $q,q',n$. We know $0\leq q\leq (u'-1)$ and $0\leq q'\leq(u-1)$ and $gcd(u,u')=1$ (along with $|u-u'|>\beta\min(u,u')$ for some $\beta\in(0,1)$).

Assuming $C(u,u')=uu'-u-u'$ and the received signal satisfies $$uq+u'q'+n< \gamma C(u,u')+\gamma'(u+u')$$ then what is the largest $\gamma,\gamma'$ we can have so that we can have recover $q,q'$ uniquely if $0\leq q\leq(u'-1)$ and $0\leq q'\leq(u-1)$ and $gcd(u,u')=1$ holds?