Define a $(7,4)$ Hamming code by the parity$-$check matrix

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

Encode the following messages $($you will need $G)$:

$($a$)1000$

$($b$)1100$

$($c$)1011$

$($d$)1001$

$($e$)1111$

Decode the following received words:

$($a$)1100001$

$($b$)1110111$

$($c$)0010001$

$($d$)0011100$