Question. If the block has only fifteen qubits and distance three, why are lookup decoding, flagged preparation, code switching, and AutDEC four different decoder claims?
A decoder consumes a specified classical record produced by a specified noise and measurement process. Changing that record changes the inference problem even when the final data block has the same stabilizer matrix.
For a Z error on the qubit labeled v, the X-check syndrome is
H_X e_v^T = v.
Because the qubit set contains every nonzero four-bit vector exactly once, the fifteen possible one-qubit Z errors give all fifteen nonzero four-bit syndromes. The syndrome is literally the binary qubit label. This is the clean Hamming-style address table inherited from the evaluation matrix.
A zero syndrome after perfect measurement means either no Z component or a higher-weight support in the X-check kernel; it does not prove the data is error-free.
An X error is detected by the ten chosen Z generators. Its syndrome is the corresponding column of H_Z. The nullspace algorithm fixes this ten-bit basis deterministically, but those ten bits do not carry an invariant four-bit geometric name: row-recombining the Z generators changes the syndrome coordinates without changing the code.
For the shipped basis, all fifteen columns are nonzero and distinct. A zero column would be an undetected weight-one X operator; equal columns would make their weight-two difference undetected. Either outcome would contradict the exact pure-X distance seven. A finite table can therefore locate every one-qubit X error. A Y error activates both tables at the same physical location.
This perfect-syndrome lookup corrects every arbitrary one-qubit Pauli error. It makes no promise for an untrusted extraction circuit or multiple data faults.
Measuring a high-weight stabilizer couples ancillas to several data qubits. An ancilla fault of the propagating Pauli type can spread through later gates and create a correlated data error. A final block syndrome alone may not say where in the circuit that fault occurred.
Different protocols preserve different side information:
The card keeps these regimes in separate decoder rows. Chamberland and Cross analyze flag-assisted logical-state preparation (paper); Anderson, Duclos-Cianci, and Poulin give an explicit switching decoder (paper); and Butt et al. retain device-motivated preparation, verification, and postselection choices in their switching comparison (paper).
Automorphism ensemble decoding runs a base belief-propagation decoder on symmetry-permuted versions of a quantum-code problem and selects among the candidate outputs. Koutsioumpas et al. compare this with BP-OSD on quantum Reed–Muller blocks under code-capacity noise and perfect syndrome (paper).
That result does not automatically decode a flagged preparation or switching circuit. A circuit-level adapter would need to define ancilla faults, measurement faults, correlations, detector construction, the supplied flag or time history, and the logical failure event. Relabeling a final perfect syndrome cannot reconstruct side information that the model discarded.
Reinforcement-learning circuit discovery is separate again: it searches for a preparation circuit, while a decoder interprets the record produced by the chosen circuit. The 2026-08-02 literature sweep found no basis for renaming the circuit-search procedure itself as the decoder.
The constructor has no size parameter and always returns the same distance-three block. An asymptotic threshold requires a sequence whose distance grows, a noise normalization, an extraction protocol, a decoder, and a fitted logical observable. A concatenation level, a growing quantum Reed–Muller family, or a complete factory family could supply such a scaling variable; the bare block cannot.
As of the 2026-08-02 decoder sweep, no fetched source supports an intrinsic asymptotic threshold for the single [[15,1,3]] lookup decoder. This packet therefore reports no threshold, pseudo-threshold, or one-point logical rate.
A comparable circuit-level row must name:
Without that tuple, a number cannot be attached honestly to this check matrix.