= Solution
Use base-two logarithms, so <Von Neumann entropy> is measured in bits. Take an <orthonormal basis> of the code and purify its maximally mixed logical state as
$$
|\Omega\rangle_{RQ}=2^{-k/2}\sum_{j=1}^{2^k}|j\rangle_R|j_L\rangle_Q.
$$
Only a $2^k$-dimensional reference support is needed; it can be embedded in the larger reference <Hilbert space> in the hint. Its <reduced density matrix> is maximally mixed on that support, so $S(R)=k$.
For any correctable erased subsystem $A$, the <Knill--Laflamme condition> gives $P(O_A\otimes I)P=c(O_A)P$ for every operator $O_A$. Taking its <matrix> elements in the logical basis shows that $\operatorname{Tr}_{Q\setminus A}|i_L\rangle\langle j_L|=0$ for $i\ne j$, and that the diagonal reductions are all the same state $\rho_A$. Thus
$$
\rho_{RA}=\rho_R\otimes\rho_A.
$$
In particular, the reference is decoupled from every set of at most $m=d-1$ erased <qubits>.
For an information-carrying code $k>0$, first justify that two disjoint sets of size $m$ fit. If $2m\geq n$, partition all the physical <qubits> into sets $A,B$ each of size at most $m$. Both are correctable. Additivity of <Von Neumann entropy> for a <product state>, and equality of complementary entropies for the <pure state> on $RAB$, give
$$
S(R)+S(A)=S(B),\qquad S(R)+S(B)=S(A).
$$
Adding yields $2S(R)=0$, contradicting $S(R)=k>0$. Hence $2m<n$.
Now choose disjoint $A,B$ of size $m$, and let $C$ contain the remaining $n-2m$ <qubits>. The state on $RABC$ is pure. Product-state entropy additivity, equality of complementary entropies, and <Subadditivity of Von Neumann entropy> give
$$
S(R)+S(A)=S(RA)=S(BC)\leq S(B)+S(C),
$$
and
$$
S(R)+S(B)=S(RB)=S(AC)\leq S(A)+S(C).
$$
Adding and cancelling proves $S(R)\leq S(C)$. Finally use the <maximum entropy of a quantum state> bound $S(C)\leq\log_2\dim\mathcal H_C=n-2m$. Therefore the <quantum Singleton bound> is
$$
\boxed{k\leq n-2(d-1),\qquad n-k\geq2(d-1).}
$$
This proof uses <quantum erasure correction>, not nondegeneracy, so it applies to degenerate codes as well. A code encoding no logical information requires a separate distance convention; the argument above concerns the usual $k>0$ coding problem.
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