Suppose $S \in \mathbb{Z}_q^{m \times m}$, and the norm of $S$ is less than an upper-bound $\beta$. i.e., the norm of $S$ is short.
Additionally, $A_1, \cdots, A_k, C_1, \cdots, C_k \in \mathbb{Z}_q^{m \times n}$.
Here, $k \geq m>n$, and $q$ is a large prime.
And $S$ is unknown.
The following equations are satisfied
$SA_1 = C_1$
$SA_2 = C_2$
$\vdots$
$SA_k = C_k$
Can I solve this problem by using Gaussian elimination or any method of system of linear equations?
Note that I think this problem is similar with the lattice hard assumption - SIS.
I am not sure whether this problem has a solution.