For the $(15,11)$ Hamming code,

$($a$)$ What is the rate of this code?

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

$($c$)$ Suppose you are given the following parity check matrix $H$ and generator matrix $G$ in systematic form.

$]]$

$111100011100100$

$110011011010010$

$[[101010110110001$

If the received sequence $r=(001000111001110),$ which received bit is in error? What is the decoded sequence?