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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' ...

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