= Solution
Interpret $t$ as the largest guaranteed radius for correcting all unknown-location <Pauli operators> of weight at most $t$ together. Products of two such errors have weight at most $2t$, so the <Knill--Laflamme condition> holds when $2t<d$. Conversely, let $F$ be a minimum-weight <Pauli operator> with nonscalar compression $PFP$. If $d\leq2t$, split its support into two sets of size at most $t$ and write $F=E_a^\dagger E_b$, up to an irrelevant phase. These two admissible errors violate the <Knill--Laflamme condition>. Thus the <correctable Pauli error radius> is
$$
\boxed{t=\left\lfloor\frac{d-1}{2}\right\rfloor,\qquad d\in\{2t+1,2t+2\}.}
$$
In particular, $d=2t+1$ requires the additional assumption that $d$ is odd. Distance is not uniquely determined by $t$ alone. Nor does the weight of a particular correctable error characterize this radius: particular higher-weight errors can also be correctable.
For a concrete even-distance example, the <stabilizer code> generated by $X^{\otimes4}$ and $Z^{\otimes4}$ has parameters $[[4,2,2]]$. Every single-qubit nonidentity <Pauli operator> anticommutes with a generator, so it has zero compression. But $Z_1Z_2$ commutes with both generators and acts nonscalarly on the code: it is unitary there and its code <trace> is zero. Hence the distance is two and its guaranteed radius is zero. This shows why an unconditional odd-distance formula would be incorrect.
Ignoring overall phases, an error of weight $j$ has $\binom nj$ choices of support and three choices, $X,Y,Z$, at each affected <qubit>. The identity is the weight-zero term, so
$$
\boxed{N(t)=\sum_{j=0}^t3^j\binom nj.}
$$
For a <nondegenerate quantum code>, distinct correctable Pauli errors have orthogonal error-image subspaces. Each has dimension $2^k$, since a <Pauli operator> is unitary. All $N(t)$ subspaces lie in the physical <Hilbert space> of dimension $2^n$. Counting dimensions proves the <quantum Hamming bound>
$$
N(t)2^k\leq2^n,
\qquad
\boxed{N(t)\leq2^{n-k}.}
$$
Degeneracy would allow distinct errors to share an error-image subspace, so this particular counting argument would then fail.
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