When does the SIS (Short Integer Solution) Lattice-problem start becoming easy (According to the parameters size)?
The problem becomes easy (as in `solvable in polynomial time') if $$\beta \geq \min_{k=1 \dots m} C^k \cdot q^{n/k}$$ for some constant $C$. This follows from:


volume $q^{n}$ for the $q$-ary kernel lattice 
a Hermite approximation factor of $C^k$ for lattice reduction algorithms (LLL/BKZ) over a lattice of dimension $k$
noting that one can `ignore columns' to work with a lattice of dimension $k \leq m$


Some more details in Section 3 of

https://homepages.cwi.nl/~dadush/teaching/lattices-2018/notes/lecture-9.pdf

(where $C=\gamma_2$ because we just consider what is provable with LLL, but all other smaller fixed constants $C&gt;1$ are also reachable in polynomial time)