Parity checks become a useful syndrome only when their order is fixed, so each bit pattern names a particular error location in the toy code. The mapping is exact only for the declared repetition word, single-bit error model, and noiseless parity checks; real hardware adds measurement error, gate faults, correlated errors, leakage, and finite-shot uncertainty.
A syndrome is useful only when the parity-bit order is fixed. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
Left parity comes first in the syndrome
After a flip at index 0, the word is 011 and the syndrome is 10.
s=10i=0
Both parity checks fire for a middle flip
After index 1 flips, the word is 101 and the syndrome is 11.