= Solution
For $r(X)=1+X+X^7$, reduction using $\alpha^4=\alpha+1$ gives
$$
(S_1,S_2,S_3,S_4)=(\alpha^3,\alpha^6,\alpha^8,\alpha^{12}).
$$
The degree-two syndrome recurrence yields
$$
\boxed{\sigma(Z)=1+\alpha^3Z+\alpha^9Z^2}.
$$
Checking the fifteen nonzero field elements shows that this polynomial has no root in $\mathbb F_{16}$. It therefore cannot be expressed as $(1-\alpha^iZ)(1-\alpha^jZ)$ 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.
Back to article page
