  In this work we study the single-shot performance of higher dimensional
hypergraph product codes decoded using belief-propagation and
ordered-statistics decoding [Panteleev and Kalachev, 2021]. We find that
decoding data qubit and syndrome measurement errors together in a single stage
leads to single-shot thresholds that greatly exceed all previously observed
single-shot thresholds for these codes. For the 3D toric code and a
phenomenological noise model, our results are consistent with a sustainable
threshold of 7.1% for $Z$ errors, compared to the threshold of 2.90% previously
found using a two-stage decoder~[Quintavalle et al., 2021]. For the 4D toric
code, for which both $X$ and $Z$ error correction is single-shot, our results
are consistent with a sustainable single-shot threshold of 4.3% which is even
higher than the threshold of 2.93% for the 2D toric code for the same noise
model but using $L$ rounds of stabiliser measurement. We also explore the
performance of balanced product and 4D hypergraph product codes which we show
lead to a reduction in qubit overhead compared the surface code for
phenomenological error rates as high as 1%.
