A block code is described by the parity check matrix indicated in Fig. $1.$

Indicate the possible code$-$words. Is this a cyclic code?

What is the probability of error in case of hard and soft decision $?$

What is the minimum required bandwidth $($in case of binary modulation$)$ if the information bit$-$rate is equal to

$10$Mbi$\frac{t}{s}.$

$H=\left[\begin{array}{ccccccc}1& 0& 1& 1& 1& 0& 0\\ 1& 1& 1& 0& 0& 1& 0\\ 0& 1& 1& 1& 0& 0& 1\end{array}\right]$

Consider the Hamming code with $N=127.$ Determine the error probability in case of both hard $($use the more

precise estimation$)$ and soft decoding.

Design a $(6,2)$ cyclic code by choosing the shortest possible generator polynomial ${?}^1.$ Determine the generator matrix

$G($in the systematic form$)$ for this code and find all the possible codewords. How many errors can be corrected by

this code?