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For what $(n,k)$ there exists a polynomial $p(x) \in F_2[x]$ s.t. $\deg(p)=k$ and $p$ divides $x^n-1$?

Motivation: if exists $p(x)$, then ideal generated by $p(x)$ is "cyclic error correcting code". It seems to me not for all $n,k$ it exists, but MatLab generates some polynoms for cyclic codes for all $n,k$ I tried. So something is wrong with my understanding.

The question probably elementary, sorry.

2 Answers 2