Consider a $(7,4)$ linear block code defined by the generator matrix G:

G $=$

$$

$$

$$

$$

$1000110$

$0100011$

$0010111$

$0001101.($a$)$ Find the parity check matrix H of the code in systematic form.$($b$)$ Find the rank of G and H matrices. Give an example of $4-$bit message encoding$$through the generator matrix G$.($c$)$ Find dmin and rate of C$.($d$)$ Suppose we add more columns to G$.$ What will be the effect on the following$$parameters of the code: code length, dimension, and rate? Compare these with$$the original parameters.