Deciding the logical effect of optimal decodes below half distance

Determine whether there exists a polynomial‑time algorithm that, given a syndrome of the two‑dimensional colour code whose minimum‑weight decoding has weight less than d/2 (where d is the code distance), decides whether that minimum‑weight decoding flips the logical state.

Background

The authors show it is NP‑hard to decide whether the minimum‑weight decoding flips the logical state for certain constructed syndromes. However, in their construction the optimal error set has weight exceeding d/2, a regime where even good decoders can legitimately fail and where hardness might be expected.

For practical error correction, the most relevant regime is when the true optimal error set has weight strictly below d/2, where an exact minimum‑weight decoder, if available, would succeed. It remains unresolved whether deciding the logical effect of the optimal decode is computationally tractable in this below‑threshold regime.