g$)$ In cyclic coding all bit patterns are expressed as polynoms in GF$(2).$ On the transmitter side

we can calculate the code word $c(x)$ by a multiplication of $c(x)=f(x)*g(x),$ and on the re$-$

ceiver side we can get back the information word by $f(x)=c\frac{x}{g}(x).$ If this division does not

not work out even, i$.$e$.$ does have a non$-$zero remainder, then the code word was corrupted

with bit error$($s$)$ and the remainder itself constitutes the syndrom. $9P$

Calculate the product: $c(x)=f(x)=g(x)$

$c(x)=(3P)$

$x^3+x+1$

$+x^3*(x^3+x+1)$

$=$

$x^3+x+1$

$+x^6+x^4+x^3$

$=$

$x^5+x^4+x+1$

$c=[1010011]$

h$)$ Verify that the Generator Matrix $G$ derived from $g(x)$ is valid and populate the symbol table

right $,$ Attention: This code is not systematic. This means you cannot see the information

word in the code word directly. You have to use either the G $-$matrix or polynom multiplication in or$-$

der to get the code word. Result see symbol table below: $($SP$)$

i$)$ Find and mark the code word $c(x)$ in the symbol table $$ see yellow marking below: $(2$P$)$