A matrix rank problem over finite fields: Is that a known problem?
When $p^q&gt;3k$, then there is an easy randomized algorithm for this problem (with error probability zero and polynomial expected running time). The algorithm prints "IMPOSSIBLE" if there is $i$ such that $B|w_i$ has more than $n-|w_i|$ rows that consist entirely of zeroes. Otherwise, the algorithm prints "POSSIBLE". It then chooses uniformly random matrices $A$ until it finds one that satisfies the requirements. Below I'll prove that a random matrix $A$ satisfies these requirements with at least constant probability.

If the algorithm printed "IMPOSSIBLE" then it is easy to see that indeed there is no $A$ that satisfies the requirements. Now, consider the case that the algorithm printed "POSSIBLE".

A known result states that a random matrix over a field of size $|F|$ is full-rank with probability $\ge 1-2/|F|$. To see this consider the columns one by one, and compute the size of the linear subspace that they have to "dodge". Work from the last column to the first. You get that the chance that the matrix is full rank is $(1-1/|F|)\cdot(1-1/|F|^2)\cdot\ldots$. In fact, the same proof can be seen to work even if some of the entries are fixed to zero, as long as no entire row is fixed to zero. QED.