Syndromes and Corrections

Parity Checks Name The Error

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=0s=10\quad i=0
Left error syndromeSyndrome order is left parity then right parity.syndrome check

Both parity checks fire for a middle flip

After index 1 flips, the word is 101 and the syndrome is 11.

s=11i=1s=11\quad i=1
Middle error syndromeThe two parity bits identify the middle carrier.syndrome check

Right parity alone names the right carrier

After index 2 flips, the syndrome is 01.

s=01i=2s=01\quad i=2
Right error syndromeThe lookup table has all single-error entries.syndrome check