Due to 5G deployment, there is significant interest in LDPC decoding. While much research is devoted on efficient hardwiring of algorithms based on Belief Propagation (BP), it has been shown that LDPC decoding can be formulated as a combinatorial optimization problem, which could benefit from significant acceleration of physical computation mechanisms such as Ising machines. This approach has so far resulted in poor performance. This paper shows that the reason is not fundamental but suboptimal hardware and formulation. A co-designed Ising machine-based system can improve speed by 3 orders of magnitude. As a result, a physical computation approach can outperform hardwiring state-of-the-art algorithms. In this paper, we show such an augmented Ising machine that is 4.4$\times$ more energy efficient than the state of the art in the literature.