Let $q$ be an odd number and $g_i = (g_{i1} g_{i2} \dots g_{ir}) \in \Bbb Z_q^r$ a list of vectors with $i\in\{1,\ldots,L\}$. Let each $g_i$ have $0 < k < r$ zero entries.
What is the maximum size of a subset $T\subseteq\Bbb Z_{q}^{r}$ such that for every pair of vectors $a,b \in T$, $a-b \neq \pm g_i$ for all $i\in\{1,\ldots,L\}$?