Binary code and Hamming distance
In a linear code, the minimum distance is the same as the minimum weight of a non-zero codeword. It's a nice exercise to prove this, and reduces the computation time quadratically.

Computing the minimum distance of a code given the generator matrix (or the parity check matrix) is NP-hard, that is, given a generator matrix (or a parity check matrix) and an integer $w$, it is NP-complete to determine whether the minimum distance of the code generated by the matrix is at most $w$. The task could become easier if the code has some structure, but for arbitrary codes it is quite difficult.