The only information on the matrix $H$ is that its rows are all the elements in $K^7$ ($K=\{0,1\}$) of odd weight (odd amount of ones). If I count them, there are 7 elements of weight 1, there are $7C3=35$ of weight 3, $7C5=21$ of weight 5 and 1 of weight 7. All in all, $64$ elements in $K^7$ of odd weight. Now the matrix $H$ is a $64\times7$ matrix.
How to show that $H$ is a parity-check matrix for some (binary) linear code $C$? The definition of a parity check matrix states it has to be $n\times(n-k)$, but since $n=7$ (codes in $K^7$ are of length $7$), this can not be true. With this I don't even know how to get the size of the code $C$ of which $H$ is the parity check matrix.
Maybe the length of the code $C$ is actually 64?