In this paper by Baranes et al., they introduce decoding hypergraph, denoted DEM_ij, extracted using gauge detectors on stim circuits that consist of: a set of hyperedges (error events) and their associated detector configurations. Each decoding hypergraph DEM_ij is a matrix, a collection of error probabilities p_n and corresponding detector configurations Dn: DEM_ij = {(D_n, p_n)}. They approximate the decoding problem of loss-induced errors by summing those decoding hypergraphs over all possib