Generator polynomial of a cyclic code ($g(X)$)

The unique monic divisor $g(X)$ of $X^n-1$ generating a cyclic code is its generator polynomial. The code has dimension $n-\mathrm{deg}g$.

Let $\alpha$ be a root of the generator polynomial of a cyclic code $g(X)$. Every codeword $c(X)$ satisfies $c(\alpha )=0$, so a received word $r=c+e$ satisfies $r(\alpha )=e(\alpha )$. For a single-symbol error $e(X)=a{X}^j$, the evaluation $r(\alpha )=a{\alpha }^j$ identifies its value and position when the relevant powers of $\alpha$ are distinct.