Use the lookup table to separate no error, one repairable error, and a failed
double-error case. The toy rows accept 111 with syndrome 00, repair 101 back to
111, and reject 001 because the repair lands on 000 instead of the original word;
real acceptance checks depend on noisy stabilizer bits, calibration, decoding
rules, finite samples, and uncertainty.
No error, one error, and two errors show what the correction table can and cannot accept. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
No-error lookup keeps the word accepted
Word 111 has syndrome 00, correction index -1, and accepted bit 1.
s=00c=−1a=1
One middle error repairs back to the code word
Error word 101 repairs to 111 with accepted bit 1.
wfixed=111a=1
Two errors can look like one but fail acceptance
Double-error word 001 repairs to 000 instead of the original word, so accepted bit 0 is checked low.