In each square of a $4 \times 4$ table a number $0$ or $1$ is written, such that the product of every two neighboring squares is $0$ (neighboring by side). $a)$ In how many ways is this possible to do if the middle $2\times 2$ is filled with $4$ zeros? $b)$ In general, in how many ways is this possible to do (regardless of the middle $2 \times 2$)?