$(1)$ The parity check matrix of the $(15,11)$ Hamming code is given as follows:

Construct the parity$-$check matrix $H$ in the case of systematic encoding.

Find the generator matrix $G$ from the above parity$-$check matrix $H$ in the case of $(15,11)$

systematic encoding.

Explain the procedure for error correction when a single$-$bit error occurs.

$H=([0,0,0,0,0,0,0,1,1,1,1,1,1,1,1],[0,0,0,1,1,1,1,0,0,0,0,1,1,1,1],[0,1,1,0,0,1,1,0,0,1,1,0,0,1,1],[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1])$