Consider the following code with message length $k=3$ and block length $n=7$ :

$($a$)$ Calculate the Hamming distance of the code.

$($b$)$ How many errors can it detect? How many errors can it correct?

$($c$)$ Encode the message $011110100.$

$($d$)$ Decode the message $001100011101111101010$ with knowing that at most $1$ error can occur in

message transmission.

$($e$)$ Show that the code is linear. What is a basis for the code?

$($f$)$ Find a generator matrix and check matrix for the linear code.