I'm getting a hard time with the following question:
Let $G$ be a generator matrix of an $(n,k,d)$ binary linear code, then $G$ has at least $kd$ ones on it.
The question is taken from Essential Coding Theory (question 1.12).
I tried to use the linear dependency of columns of $G$ and the distance between them. I also tried to assume towards contradiction that $G$ contains at most $kd-1$ ones, but didn't get a contradiction.
Any hint will be much appreciated. Thanks.
In the generator matrix $G$ has $k$ rows because the binary code $C$ has papmeter $[n, k, d]$. In $G$ every row is a non zero codeword of $C$. Since $C$ is $[n, k, d]$ binary code, every non zero codeword has $\geq d$ ones. So total number of ones in $G$ is $\geq dk$.