= Solution
If every $\operatorname{wt}(E_a)<d/2$, then
$$
\operatorname{wt}(E_a^\dagger E_b)
\leq\operatorname{wt}(E_a)+\operatorname{wt}(E_b)<d.
$$
By the definition of the <distance of a stabilizer code>, no Pauli operator of weight below $d$ is a nontrivial logical operator. Part (b) therefore proves the <local correctability of a stabilizer code> for this error set.
The converse fails because pairwise products, rather than individual weights, control correctability. For example, let $P$ be a high-weight Pauli outside $C(\mathcal S)$ and take the error set $\{I,P\}$. If $P$ anticommutes with a stabilizer, then $\Pi P\Pi=0$, while $P^\dagger P=I$; the KL conditions hold even when $\operatorname{wt}(P)\geq d/2$. A still simpler singleton set containing any known unitary Pauli error is always reversible regardless of its weight.
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