= Solution
The <polynomials> are over $\mathbb F_2$, and multiplication gives $gh=X^7+1=X^7-1$. A <cyclic code> of length seven generated by a degree-three <polynomial> has dimension $7-3=4$: the multiples $ag$ with $\deg a<4$ are distinct representatives, so its rank is \b[four].
The root $\alpha$ has order seven in $\mathbb F_8^\times$: it is neither zero nor one, and the multiplicative group has prime order seven. A weight-one word cannot vanish at $\alpha$. A weight-two word $X^i+X^j$, with distinct indices between zero and six, cannot vanish either, because that would give $\alpha^{i-j}=1$. Every codeword vanishes at $\alpha$, so its nonzero weight is at least three. Since $g$ itself has weight three,
$$
\boxed{d_{\min}=3.}
$$
From $\alpha^3=\alpha+1$ obtain $\alpha^4=\alpha^2+\alpha$ and $\alpha^5=\alpha^2+\alpha+1$. Therefore $r(\alpha)=\alpha^5+\alpha^3+\alpha=\alpha^2+\alpha=\alpha^4$. The one-bit error $X^4$ has precisely this <syndrome>, so correcting it gives
$$
\boxed{c(X)=X^5+X^4+X^3+X=g(X)(X^2+X).}
$$
Its distance from the received word is one. The <minimum distance> three makes this nearest codeword unique: two codewords each at distance one from the received word would be at distance at most two from each other.
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