Exact encoded-state failure with a faulty parity check ($P_{\mathrm{fail}}=q+(1-q)(3p^2-2p^3)$)

For independent channel bit flips of probability $p$ and the specified extraction fault of probability $q$, require recovery of an arbitrary input in the original encoded subspace. Every extraction-fault event leaves nonzero error syndrome; without that fault, channel errors are corrected exactly at weight zero or one. Thus success probability is $(1-q)[(1-p{)}^3+3p(1-p{)}^2]$. Compared with a single unencoded bit-flip channel, strict improvement for $0<p<1/2$ requires $q<p(1-2p)/[(1-p)(1+2p)]$. A different decoder or an added later ideal recovery changes the performance criterion and must be specified separately.