5-qubit [[5,1,3]]

The parameters n=5, k=1, d=3 give rate 1/5 and meet the minimum-length bound for correcting one arbitrary qubit error. Four independent weight-four checks yield sixteen syndromes; finite-length optimality is not an asymptotic-rate or LDPC claim.

The code has useful cyclic and Clifford symmetries, but transversal gates alone do not supply a universal logical gate set. Non-Clifford computation generally needs code switching, magic states, or a verified encoded-gate sequence.

Under an ideal Pauli model, a four-bit lookup table maps each nonzero syndrome to a minimum-weight representative. Circuit-level decoding must include measurement faults and propagation in a spacetime syndrome rather than reuse the noiseless fifteen-entry table.

The measurement order of weight-four mixed-Pauli checks controls the direction of hook errors. Flag ancillas or verified ancillas can expose dangerous propagation, but fault tolerance belongs to a specific circuit, not automatically to the [[5,1,3]] parameters.

The five-qubit code is a common small-processor benchmark for full encoding, injected errors, and recovery because it minimizes the number of data qubits. Experimental comparisons must still count ancillas, repeated syndrome rounds, leakage, and postselection.

Like the Steane and Shor codes it is a distance-three one-logical-qubit block code, but it is shorter and non-CSS. Equal distance specifies only the minimum logical weight, not the syndrome structure, gate set, or fault-tolerant overhead.

“Perfect” denotes bound saturation, not best performance under every physical noise model. Bias, connectivity, and measurement cost can reverse comparisons; experimental claims should state whether decoding is ideal, single-shot, or based on repeated fault-tolerant rounds.