Least non-zero multiple of solution to linear equation system that is integer

Question

Given a system of equations on the form $A\bar{x}=\bar{b}$, where $A\in\{-L,-L+1,\dots,L-1,L\}^{d\times d}$ (I don't know if this is the correct notation) has full rang and $\bar{b}\in\{-L,-L+1,\dots,L-1,L\}^d\setminus\bar{0}$, can we say anything about the non-zero multiple of $\bar{x}$ with the smallest magnitude? Can we find any upper bound on its magnitude somehow?