The Quantum Singleton Bound states that for an error-correcting code with $n$ physical qubits and $k$ encoded qudits, and some subsystem $R$ of $m$ qudits that can 'access the entire quantum code', it is necessary that $m \ge \frac{n+k}{2}$.
As I understand (from section 4.3 of Harlow's TASI notes), one way to state the condition for 'accessing the entire code' is the Knill-Laflamme condition, which is the following.
Let $\bar{R}$ denote the complement of $R$, $\mathscr{H}_\bar{R}$ be the space of operators supported on $\bar{R}$, and $P$ denote the projection matrix onto the code subspace $\mathscr{H}_{code}$. Then for any operator $O_{\bar{R}} \in \mathscr{H}_\bar{R}$, $P O_{\bar{R}} P = \lambda P$, where $\lambda$ is some constant that depends on the operator $O_{\bar{R}}$
This means that operator supported on the complement region $\bar{R}$ has no effect on measurements on $\mathscr{H}_{code}$.
I'm confused because this does not seem compatible with the toric code. Because, it can be shown that in the toric code (where the number of encoded bits $k=2$), the Knill-Laflamme condition is satisfied for $\bar{R}$ being any contractible region of qubits, i.e. for $R$ containing the union of two distinct nontrivial cycles on the torus. In this case on a torus of length $L \times L$, we will have the number of physical qubits being $n = 2 L^2$ and the number of bits needed to access being $m = 4 L$. So, it seems that the singleton bound is explicitly violated.
Where does the logic I'm presenting fail, and why should the Toric Code be compatible with this?
EDIT: I think I found the issue. It turns out that Harlow was a bit unclear in what he meant in his statement that some subsystem of $m$ qudits being able to access the information. He didn't actually mean that, but meant a different statement, which I'll summarize in the following theorem.
Theorem: Suppose that in a qudit system S of size $n$, there are subsystems A,B both of size $m$ such that their complements $A^c$ and $B^c$ (both of size $n-m$) are disjoint and can be erased (not necessarily simultaneously) and the errors be recovered (in the Knill-Laflamme sense above, or in some other equivalent senses as in Theorem 4.1 of his TASI notes). Then, $m \ge \frac{n+k}{2}$.
Note that this condition does NOT mean the same thing as any subsystem being able to access the whole system. Rather, it means that if two disjoint regions of sizes $(n-m)$ (so that their total size is $2(n-m)$) can be erased and corrected (not necessarily simultaneously), then $m \ge \frac{n+k}{2}$.
This is equivalent to the statement that Preskill made in his notes (see Norbert's reply below), since $n-m = d-1$ relates $n-m$ to the distance $d$ of the code, which means that $n-k \ge 2(d-1)$, which is indeed the Singleton bound.
So long story short, the above bound $m \ge \frac{n+k}{2}$ does not hold for generally being able to access the logical operators from a subsystem of size $m$.