hiw is thb$)$ A single error$-$correcting BCH code is constructed over $F$ using the primitive element $.$ c$)$ Using the same BCH code, an encoded message is received with corresponding received polynomial

$D(x)=x^{11}+x^{12}+x^{13}+x^{14}$

It is known that exactly one error occurred in the transmission.

This error must have occurred at the coefficient of $x^k$ in $D(x),$ where $k$ is given by:

$k=$

Suppose we wish to use this to encode the message $(0,1,1,1,0,1,0,0,0,0,0).$

Then the encoded message as a codeword polynomial is

$C(x)=R(x)+x^5+x^6+x^7+x^9$

where the polynomial $R(x)$ is given by:

$R(x)=$