Let's consider $(7,4)$ Hamming code.

$(1)$ Generator matrix has the type of $G=[I_k|P]$ when Parity check matrix $H$ has the type

of $H=[P^T|I_{n-k}]$ in a systematic block code. When the Parity check matrix of $(7,4)$

Hamming code has the following matrix, find the Generator matrix G$.$

$(2)$ When the input message is $"1100",$ find the codeword using the above Generator matrix.

$(3)$ When the received code vector is $"1011000",$ determine whether there is any error in

the received code vector and if there is any error, then find the right codeword

transmitted by the sender$.$