= Hamming code as a cyclic code
{c}
Let $\omega$ generate $\mathbb F_{2^s}^{\times}$, $s\geq2$, and set $n=2^s-1$. The binary <cyclic code> defined by $c(\omega)=0$ has a <parity-check matrix> consisting of every nonzero vector of $\mathbb F_2^s$ once, after choosing a <basis> and ordering its columns. It is therefore a <Hamming code>. Its <generator polynomial> is the <minimal polynomial of an algebraic element> $m_\omega(x)=\prod_{j=0}^{s-1}(x-\omega^{2^j})$, of degree $s$.
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