Ternary Hamming code as a cyclic code ($g(X)={\prod }_{j=0}^{s-1}(X-{\alpha }^{3^j})$)

For odd $s\ge 3$, put $n=(3^s-1)/2$ and choose $\alpha$ of order $n$ in ${\mathbb{F}}_{3^s}$. Because $n$ is odd, the powers ${\alpha }^i$, $0\le i<n$, represent every projective point once. The cyclic code defined by $c(\alpha )=0$ is therefore monomially equivalent to a ternary Hamming code, with the displayed generator polynomial of a cyclic code. This statement requires a qualification: at $s=2$, every ternary cyclic length-four, dimension-two code has a weight-two generator, so none has Hamming distance three.