Question $3$

Consider a $(7,4)$ code whose generator matrix $G$ is given by:

$G=\left[\begin{array}{ccccccc}1& 0& 1& 0& 1& 0& 1\\ 0& 1& 1& 0& 0& 1& 1\\ 0& 0& 0& 1& 1& 1& 0\\ 1& 1& 1& 0& 0& 0& 1\end{array}\right]$

$(1)$ Derive $H,$ the parity$-$check matrix of the code.

$(2)$ What is the error$-$correcting capability of the code?

$(3)$ What is the error$-$detecting capability of the code?

$(4)$ Compute the syndrome for the received vector $1011001.$ Is this a valid code vector?