Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 3 / 3H / Solution

The relation $g(X)h(X)=X^7-1$ makes this a length-seven binary cyclic code. Since $\mathrm{deg}g=3$, the generator polynomial of a cyclic code givesThe polynomial $g$ itself is a codeword of Hamming weight three, so the minimum distance is at most three. A nonzero word of weight one cannot vanish at the nonzero element $\alpha$. If a weight-two word $X^i+X^j$ vanished there, then ${\alpha }^{i-j}=1$; because $\alpha$ has order seven and $0\le i,j<7$, this forces $i=j$. Hence there are no nonzero words of weight below three, and

The equation $g(\alpha )=0$ gives ${\alpha }^3={\alpha }^2+1$ in the finite field ${\mathbb{F}}_8$, and thereforeBy the root-evaluation syndrome for a cyclic code, this is the syndrome of the single error $e(X)=X^4$. Since $d(C)=3$, minimum-distance decoding uniquely corrects one error. Subtracting and adding agree over ${\mathbb{F}}_2$, so the decoded codeword is