I am working on questions about coding theory. The set C is the set of all words in binary code that have an even weight and are of length n.
I have already proven that this is a linear code. Now I have to determine the dimension and basis of C. However, I cannot seem to wrap my head around this problem. Can anyone help me with this?
This is the set of all even paritys. So you can creat any message as long as the last bit is the sum of previous bits. $H = \begin{bmatrix} 1_{n\times1}\end{bmatrix}$. And if you solve $GH^T = 0$, the basis will be $G=\begin{bmatrix} I_{n-1} & 1_{(n-1)\times1}\end{bmatrix}$. So the dimension as mentioned is $n-1$.