  Quantum sensors are expected to be a prominent use-case of quantum
technologies, but in practice, noise easily degrades their performance. Quantum
sensors can for instance be afflicted with erasure errors. Here, we consider
using quantum probe states with a structure that corresponds to classical
$[n,k,d]$ binary block codes of minimum distance $d \geq t+1$. We obtain bounds
on the ultimate precision that these probe states can give for estimating the
unknown magnitude of a classical field after at most $t$ qubits of the quantum
probe state are erased. We show that the quantum Fisher information is
proportional to the variances of the weight distributions of the corresponding
$2^t$ shortened codes. If the shortened codes of a fixed code with $d \geq t+1$
have a non-trivial weight distribution, then the probe states obtained by
concatenating this code with repetition codes of increasing length enable
asymptotically optimal field-sensing that passively tolerates up to $t$ erasure
errors.
