  This paper proposes to use graph neural networks (GNNs) for equalization,
that can also be used to perform joint equalization and decoding (JED). For
equalization, the GNN is build upon the factor graph representations of the
channel, while for JED, the factor graph is expanded by the Tanner graph of the
parity-check matrix (PCM) of the channel code, sharing the variable nodes
(VNs). A particularly advantageous property of the GNN is the robustness
against cycles in the factor graphs which is the main problem for belief
propagation (BP)-based equalization. As a result of having a fully deep
learning-based receiver, joint optimization instead of individual optimization
of the components is enabled, so-called end-to-end learning. Furthermore, we
propose a parallel flooding schedule that further reduces the latency, which
turns out to improve also the error correcting performance. The proposed
approach is analyzed and compared to state-of-the-art baselines in terms of
error correcting capability and latency. At a fixed low latency, the flooding
GNN for JED demonstrates a gain of 2.25 dB in bit error rate (BER) compared to
an iterative Bahl--Cock--Jelinek--Raviv (BCJR)-BP baseline.
