The no-three-in-line problem on a torus

Let $T(\Z_m \times \Z_n)$ denote the maximal number of points that can be placed on an $m \times n$ discrete torus with "no three in a line," meaning no three in a coset of a cyclic subgroup of $\Z_m \times \Z_n$. By proving upper bounds and providing explicit constructions, for distinct primes $p$ and $q$, we show that $T(\Z_p \times \Z_{p^2}) = 2p$ and $T(\Z_p \times \Z_{pq}) = p+1$. Via Gröbner bases, we compute $T(\Z_m \times \Z_n)$ for $2 \leq m \leq 7$ and $2 \leq n \leq 19$.

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