= Systematic polynomial encoding of a cyclic code
{title2=$c(x)=x^{n-k}M(x)-(x^{n-k}M(x)\bmod g(x))$}
For a length-$n$, dimension-$k$ <cyclic code> with <generator polynomial> $g$ of degree $n-k$, let the message <polynomial> satisfy $\deg M<k$. Subtracting the remainder upon <polynomial division> by $g$ makes the displayed word divisible by $g$. The remainder has degree less than $n-k$, so the highest $k$ coordinates retain the coefficients of $M$. This gives <systematic encoding> without losing any message information.
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