Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 11K / b / iii

For $r(X)=1+X+X^7$, reduction using ${\alpha }^4=\alpha +1$ givesThe degree-two syndrome recurrence yieldsChecking the fifteen nonzero field elements shows that this polynomial has no root in ${\mathbb{F}}_{16}$. It therefore cannot be expressed as $(1-{\alpha }^{i}Z)(1-{\alpha }^{j}Z)$ for error positions $i,j$. The received word lies outside the code's two-error decoding radius, so the error positions cannot be determined uniquely by this decoder.