Ternary Hamming code ($[(3^s-1)/2,(3^s-1)/2-s,3{]}_3$)

For $s\ge 2$, use one nonzero representative from each one-dimensional subspace of ${\mathbb{F}}_3^s$ as a parity-check matrix column. The rank is $s$; no zero or proportional columns imply minimum Hamming distance at least three. Two independent columns and the representative of their sum give a weight-three relation, proving equality. Every nonzero syndrome uniquely identifies one column and one nonzero error value.

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.

Set $n=(3^s-1)/2$ and let $\beta$ be a primitive element of a finite field ${\mathbb{F}}_{3^s}$. The powers $1,\beta ,\dots ,{\beta }^{n-1}$ represent every projective point once. Their kernel code is a ternary Hamming code. Since ${\beta }^n=-1$, its polynomial description is a negacyclic code, generated by the minimal polynomial of an algebraic element $\beta$ over ${\mathbb{F}}_3$.