Redundant Code Words

One Flip Changes A Word

An indexed physical flip changes one carrier in the code word while leaving the logical-code rule available for later checks. The word update is exact only for the stated starting word, flip index, and ideal binary error model; hardware faults can be analog, correlated, mislocated, or hidden by measurement error.

A modeled error is an indexed bit flip in a checked physical word. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

The starting word is checked before the error

The protected word begins as 111.

wbefore=111w_{\mathrm{before}}=111
Before the flipThe before word is part of the check.syndrome check

One indexed bit flip changes exactly one carrier

A flip at index 1 changes the word to 101.

i=1wafter=101i=1\quad w_{\mathrm{after}}=101
Indexed flipThe error index is structural, not a caption.syndrome check