I'm reading this paper about decoding the [23,11,7] Golay code (Also QR code) generated by the polynomial $g(x)=x^{11}+x^9+x^7+x^6+x^5+x+1$, it says that if $\alpha$ is root of $g(x)$ and the associated cyclotomic set $B=\{1,2,3,4,6,8,9,12,13,16,18\}$ shows that $g(x)$ has roots $\alpha,\alpha^3,\alpha^9$. Why these are roots of $g(x)$ and not others? Also why the associated cyclotomic set is the same as the nonzero quadratic residues modulo $23$?
And why in this paper the author takes only the odd syndromes?
Thank you all.
Due to conjugacy constraints (since $g(x)$ has binary coefficients) if $z$ is a root so are the elements of $\{z^{2^i}: i\geq 0\}$ so you pick up all the elements of the cyclotomic coset as a root, if any of its elements is a root.
Quadratic residues property is also preserved by squaring. Clearly $\alpha^2$ is a QR and thus so are $\alpha^4,\alpha^8,\cdots.$
Expand your last question in the body of the post, I didn’t click on the link. That way others reading the question may benefit.