Finding dual code with Block code

Question

If there is a Block code $C$, $$ C =\{(0000),(0101),(0001),(0100)\} $$ ($4$ tuple & dimension=$2$)

How can I find dual code of Block code $C$?

Answer 1

Well, its a linear code of dimension $2$ generated by the vector $0100$ and $0001$.

So the dual code has also dimension $2$. Just find two linearly independent vectors in the dual code, such as $1000$ and $0010$. The code is thus $0000$, $1000$, $1010$ and $0010$.

You can check this by taking the inner product $\langle u,v\rangle = \sum_i u_iv_i$.