Consider the BCH 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$).$

The code is extended adding a nal parity check bit $($imposing an even number of $"1").$

$\{($a$)$ Determine the new probability of error $($in case of hard and soft decision$).$

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

to $1$ Mbit$/$s$.$

Consider the following possible codewords.

$\{($c$)00000000000001000111110101111001$ is a valid codeword $?$

$\{($d$)00000000000001111010111110001001$ is a valid codeword $?$