Consider the linear block code of length N $=31$ and generator polynomial $($in octal description$)$

$107657.$ In this code there is $1$ word composed by all zeros, $155$ words with $7$ ones, $465$ with $8$

ones, $5208$ with $11$ ones, $...$

What is the generator polynomial of this code $($g$($D$)=...)?$ Determine the number of

possible codewords, and the probability of error $($in case of hard and soft decision, using

the more precise estimation$).$

The code is now extended adding a nal parity check bit $($imposing an EVEN number of

$"1").$ Determine the new probability of error $($in case of hard and soft decision, using the

more precise estimation$).$

Consider the following possible codewords $($of the extended code$).$

$\{00000000000001000111110101111001$ is a valid codeword $?$

$\{00000000000001111010111110001001$ is a valid codeword $?$