I am studying Information theory, coding theory in particular at the moment, and I am having trouble determining how many different cycles are defined by a certain generator polinomial?
Given a polinomial, for example $g(p)=p^6+p^3+1$, and the length of the code $n = 9$ how can I determine the number of different cycles that are defined?
I know that the cycles (001)(010)(100) and (011)(110)(101) are defined, but how can I find the rest of them?
I also wonder if there is an explicit formula that can return me the result, maybe based on the different powers in the generating polinomial?
Thank you for helping me.
HINT: Every cyclic code is in particular linear and hence it has a generating matrix. There is a well-known procedure how to calculate the generating matrix or parity check matrix from the generating polynomial. Once you do this, your code is just the row space of the generating matrix or the kernel of the parity check matrix.