$1-$ Consider the $(7,4)$ Hamming code defined by the generator polynomial

$g(x)=1+x+x^3$

The codeword $0111001$ is sent over a noisy channel, producing the received word $01010$ a single error. Determine the syndrome polynomial $s(x)$ for this received word, and sho identical to the error polynomial $e(x).$

$2-$ The generator polynomial of a $(15,11)$ Hamming code is defined by

$g(x)=1+x+x^4$

Develop the encoder and syndrome calculator for this code, using a systematic form for $t1$