Please answer $10$ and $11.$ A matrix $H$ with kernel $V$ is called a parity check matrix. Applying any row operation to $H$ will not change the kernel, so the parity check matrix is not unique.

A larger code. We now consider a code with $N=32,p=2$ and $n=9.$ This code has generator matrix and parity check matrix:

$G_{\text{b}\text{i}\text{g}}=\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\\ 1& 1& 0& 0& 1\\ 1& 1& 1& 0& 0\\ 1& 0& 1& 1& 0\\ 1& 0& 0& 1& 1\end{array}\right],H_{\text{b}\text{i}\text{g}}=\left[\begin{array}{ccccccccc}1& 1& 0& 0& 1& 1& 0& 0& 0\\ 1& 1& 1& 0& 0& 0& 1& 0& 0\\ 1& 0& 1& 1& 0& 0& 0& 1& 0\\ 1& 0& 0& 1& 1& 0& 0& 0& 1\end{array}\right].$

Problem $9$ Explain how to use $H_{\text{b}\text{i}\text{g}}$ to check whether or not a vector is in the image of $G_{\text{b}\text{i}\text{g}}.$ For example, of the three vectors $100110101,101100010$ and $110111101,$ exactly one is in the image of $G_{\text{b}\text{i}\text{g}\text{.}}.$ Which one and why?

Problem $10$ Let vec$(y)$ be the vector which you just discovered is in the image of $G_{\text{b}\text{i}\text{g}}.$ Find the vector vec$(x)$ for which vec$(y)=G_{\text{b}\text{i}\text{g}}$vec$(x).($Hint: Look at the top $5$ rows of $G_{\text{b}\text{i}\text{g}}.)$

Problem $11$ Suppose that Alice meant to send thet vector $G_{\text{b}\text{i}\text{g}}$vec$(x).$ However, the first bit of her message got flipped, so Bob received vec$(y)$ instead. What will $H_{\text{b}\text{i}\text{g}}$vec$(y)$ be$?$ What would $H_{\text{b}\text{i}\text{g}}$vec$(y)$ be if the ninth bit had been flipped instead?