relevant primitive for the construction of other codes such as LT or raptor codes [@kumar; @LT-shokro]. Note that this paper will consider such a code at a vanishing rate, namely $c/\log(n)$, and determine for which values of $c$ the successful decoding of this code is still possible. Somehow unexpectedly, the Shannon capacity will also arise in this regime as shown in our main results.

-   [**Constraint satisfaction problems:**]{} is a particular case of the graphical channel studied in [@random] in the context of hypergraphs. This class of models allows in particular to recover instances of planted constraint satisfaction problems (CSPs) by choosing uniform kernels $Q$, where the vertex-variables represent the planted assignment and the edge-variables represent the clauses. In the case of a simple graph and not a hypergraph, this provides a model for planted formulae such as 2-XORSAT (model ).

-   [**Synchronization:**]{} Equation results also from the synchronization problem studied in [@ASinger_2011_angsync; @Bandeira_Singer_Spielman_OdCheeger; @Wang_RobustSynchronization; @Alexeev_PhaseRetrievalPolarization; @boumal], if the dimension is one (e.g., when each vertex-variable is the 1-bit quantization of the reflection of a signal). The goal in synchronization over $O(r)$, the group of orthogonal matrices[^2] of size $r\times r$, is to recover the original values of the node-variables 