How many solution there are finite field linear system?

Question

Let be $S$ a system of N linear equations, with K unknowns; and K > N, where the equations are linear and over $\mathbb{F}_2$, how many solutions are in this finite field linear system?

Answer 1

As in all fields, if the matrix of the system and the augmented matrix have the same rank $r\quad (r\le N)$, the set of solutions is an affine subspace of $\mathbf F_2^K$ of codimension r, hence there are $2^{N-r}$ points in this subspace. If the rank of the augmented matrix is greater than the rank of the matrix of the system, there is no solution.