Take $A,B,C,D$ pairwise coprime with $$n<A,B,C,D<2n$$ $$ n/8<|A−B|,|C−D|,|A−C|,|A−D|,|B−C|,|B−D|<n/4$$ and consider the space spanned by $3\times 4$ matrix $$N=\begin{bmatrix} -D&C&0&0\\ -B&0&A&0\\ BD&-BC&-AD&AC \end{bmatrix}.$$

Denote the $\Bbb Q$-linear space spanned by rows of $N$ by $T_{A,B,C,D}\subseteq\Bbb Q^4$ and denote the set of non-zero $\Bbb Z$ vectors in $T_{A,B,C,D}$ by $T_{A,B,C,D}[\Bbb Z]^\star$.

1. Is there a name for the quantity $\min_{v\in T_{A,B,C,D}[\Bbb Z]^\star}\|v\|_\infty$ where $\|v\|_\infty$ is largest coordinate by magnitude of vector $v$?

2. How is $\min_{v\in T_{A,B,C,D}[\Bbb Z]^\star}\|v\|_\infty$ distributed as a function of $A,B,C,D$ chosen with the constraints above and what is its average value?