The same syndrome table that repairs one physical flip can misidentify a two-error word, exposing the limit of this repetition-code check. The failure is exact for the declared double-error pattern and single-error lookup rule; real devices require distance, repeated rounds, and error-model assumptions to bound correlated faults and measurement noise.
The same single-error lookup must reject a two-error repair claim. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
A double error can imitate a single-error row
The double-error word 001 has syndrome 01, which points to correction index 2.
s=01c=2
The corrected word does not match the original
Applying that correction gives 000, so the checked acceptance bit is 0.