Got this assignment from coding class and would be very thankful for checking if my solutions are correct.
a) Find all idempotents modulo $1 + x^{17}$ of degree at most $15$
So first i find $r$ from $1=2^r mod17$ so $r=8$.
Next step is to find idempotents by partitioning $Zn$ into classes: $C_0 = \{ 0 \}; C_1 = \{ 1,2,4,16,15,13,9 \}; C_3 = \{ 3,5,6,7,10,11,12,14 \}$
Here I don't know if I've done this step correct because $C_2$ was just like $C_1$ but missing the $1$ in the set. Should i treat it as another class?
The rest of the sets are the same, so i assume there is some pattern for finding all of them. them (further there are just multipliers[sry I'm neither native english nor a math student so dont know if this term is used correctly, but I hope You know what I mean]) Are there usually only 3 such sets?
From the obtained sets i got idempotents for a which are at most degree $15$: $\{1, x^3+x^5+x^6+x^7+x^{10}+x^{11}+x^{12}+x^{14}, 1+x^3+x^5+x^6+x^7+x^{10}+x^{11}+x^{12}+x^{14}\}$
b) How many cyclic linear codes (other than $\{00...0\}$ and $K^n$) are there if $n = 17$? And if $n = 136$?
So the number of cyclic linear codes is equal to the number of idempotents? for $n=17$ it was 7, so 7 linear codes. Don't know if the formula i've found for calculating this was correct but seems right $2^{n}-1$ where $n$ is the number of sets obtained? How to find for $136$?
c)Let $I(x)$ be the idempotent in (a) in which the constant term equals zero and the coeficient of $x^ 3$ is non-zero. Find the generator polynomial of the smallest cyclic linear code of length 17 containing $I(x)$. What is the dimension of that code?
I just obtained the $GCD(1+x^{17}, x^3+x^5+x^6+x^7+x^{10}+x^{11}+x^{12}+x^{14})$, which is the generator polynomial for the cyclic linear code. Not sure about the dimension? Is it equal to $n-r$?