For a $(6,3)$ systematic linear block code, the three parity$-$check digits $c_4,c_5,$ and $c_6$ are

$c_4=d_1+d_2+d_3$

$c_5=d_1+d_2$

$c_6=d_1+d_3$

$($a$)$ Construct the appropriate generator matrix for this code.

$($b$)$ Construct the code generated by this matrix.

$($c$)$ Determine the error correcting capabilities of this code.

$($d$)$ Prepare a suitable decoding table.

$($e$)$ Decode the following received words: $101100,000110,101010.$