Problem $2$ :

Consider the linear block codes, word code length $n=7.$

$($a$)$ How many cyclic codes can be constructed which have this word length? Find their generator polynomials.

$($b$)$ For one of the possible generator polynomials find all code words and code weight spectrum. Comment the code possibilities concerning the error correction.

$($c$)$ Whether $g(x)=1+x^3+x^4$ can be the generator polynomial of cyclic $(7,3)$ code?

$($d$)$ Form a systematic generator matrix for the code with the generator polynomial $g(x)=1+x$