Extended Hamming code to cyclic code

Question

Is there any way to present [8, 4] extended Hamming code as a cyclic code? Empirically, it seems not possible; however, I cannot prove or disprove it.

Answer 1

Over $\mathbb{F}_2$, the factorization of $X^8 - 1 = X^8 + 1$ into monic irreducible polynomials is $(X+1)^8$ (by the "freshman's dream"). Thus, there is only a single cyclic binary $[8,4]$ code $C$, with the generator polynomial $(X+1)^4 = X^4 + 1$. This polynomial has Hamming weight $2$, so the minimum distance of $C$ is at most $2$. Therefore, $C$ cannot be isomorphic to the $[8,4,4]$ extended Hamming code.