Let $C$ be a $[n,k,d]_2$ code where $d$ is odd. It is known that you can construct a $[n+1,k,d+1]_2$ code by adding a column $\boldsymbol{c}_{n+1}$ to the codebook matrix where each element contains the parity of the codewords of $C$. I.e. for all codewords $[x^i_1, x^i_2, \dots, x^i_n] \in C$, we append the codebit $\oplus_{j=1}^n x^i_j$.
Is it ever possible to extend $C$ with a column containing anything other than the parities of the original codewords and increment the minimum distance by one? Why or why not?