I have the following problem I am struggling to solve.
Let $C$ be a binary $[n,k,d]$ code and let $\mathbf{c}$ be a minimum weight codeword of $C$. Let $\overline{C}$ be the punctured code obtained by $C$, puncturing on the positions $i$ where $c_i \neq 0$. I have to find the dimension of $\overline{C}$ and prove that its minimum distance is at least $\lceil \frac{d}{2} \rceil$. I am struggling with this exercise, and in general with finding the parameters of punctured code. Any help is very much appreciated :)