better alternatives to the surface code. For example, dramatically improved thresholds could be possible using concatenated codes if they enjoyed the same level of optimization as the surface code has in recent years [@Poulin06; @PhysRevA.83.020302]. Another enticing alternative is to find and use efficiently-decodable low density parity check (LDPC) codes with high rate [@Gallager1960; @LDPC13; @TZLDPC14] in a low-overhead FTEC protocol [@Gottesman13LDPC]. For these and other reasons, it is important to have general FTEC schemes applicable to a wide range of codes and to develop new schemes.

Shor EC can be applied to any stabilizer code, but typically requires more syndrome measurement repetitions than Steane and Knill EC. Furthermore, all weight-$w$ stabilizer generators are measured sequentially using $w$-qubit verified cat states. On the other hand, Steane EC has higher thresholds than Shor EC and has the advantage that all Clifford gates are applied transversally during the protocol. However, Steane EC is only applicable to CSS [@CS96; @Steane97] codes and uses a verified logical ${|+\rangle}$ state encoded in the same code to simultaneously obtain all $X$-type syndromes, using transversal measurement (similarly for $Z$). Knill EC can also be applied to any stabilizer code but requires two additional ancilla code blocks (encoded 