$($b$)$ Consider a $(7,4)$ Hamming error correction block code whose generator

matrix is

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

$($i$)$ Is this code in systematic form? Evaluate the codewords of messages

$[0011]$ and $[0010].$

$($ii$)$ Find $H,$ the parity check matrix of the code, and explain how decoding

is performed.

$($iii$)$ What is the syndrome of a received word? Compute the syndrome of

the received vector $r=[1001100].$ Is this a valid vector?