A $(7,4)$ cyclic linear block code is described by the generator polinomial g$($D$)=$ D$3+$D$2+1.$ Indicate the possible

code$-$words. Determine the generator matrix of this code, in its systematic shape.

Indicate the values assumed by the "sindrome" associated to a possible single bit error, a possible two bits error, a

possible three bits error.

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

precise estimation$)$ and soft decoding.