Let $C$ be a linear code with minimum distance $2k$. I want to show that there is a coset of $C$ that contains at least two vectors of weight $k$.
Firstly, it holds that the minimum distance of the code is equal to the lowest non-zero weight of a codeword. So this means that the weights of the codewords are greater than these of the elements of the coset of $C$, right?
Solving the system $Hx=0$, where $H$ is the parity matrix, we can find the coset of $C$, right?
But in this case we don't have a generator matrix of the dual code. How can we get information about the parity matrix?
Or how else can we get information about the coset of $C$?