Systematic polynomial encoding of a cyclic code ($c(x)=x^{n-k}M(x)-(x^{n-k}M(x)\mathrm{mod}g(x))$)

For a length-$n$, dimension-$k$ cyclic code with generator polynomial $g$ of degree $n-k$, let the message polynomial satisfy $\mathrm{deg}M<k$. 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.