\} \\
\{ E_{\text{min}}(s) \} \text{ if above set empty. }
\end{cases}
\label{eq:GeneralLookupTable}\end{aligned}$$

By $E \in \mathcal{E}_{m}(g_{i_{1}},\cdots , g_{i_{k}}) \times \mathcal{E}_{t-m}$, we are considering the set consisting of products between errors caused by $k$ flags and any error of weight $t-m$.

As will be seen below, the correction set will form a critical part of the protocol by specifying the correction applied based on the measured syndrome and flag outcomes over multiple syndrome measurement rounds. In the case where $k$ $t$-flag circuits flagged caused by $k \le m \le t$ faults, the correction applied to the data block will correspond to an element of $\mathcal{E}_{m}(g_{i_{1}},\cdots , g_{i_{k}}) \times \mathcal{E}_{t-m}$ if the measured syndrome corresponds to an element in this set (there could also be $t-m$ faults which did not give rise to a flag). However in practice, there could be more than $t$ faults and so the measured syndrome may not be consistent with any element of the set $\mathcal{E}_{m}(g_{i_{1}},\cdots , g_{i_{k}}) \times \mathcal{E}_{t-m}$. In this case, and for the error correction protocol to satisfy the second criteria of \[Def:FaultTolerantDef\], the correction will correspond to $E_{\text{min}}(s)$. These features are all included in the set $\tilde{E}_{t}^{m}(g_{i_{1}},\cdots , g_{i_{k}},s)$.

Consider a stabilizer 