Hyper Neural Network as the Diffeomorphic Domain for Short Code Soft Decision Beyound Belief Propagation Decoding Problem

We proposed topological interpretation of the Tanner-Forney-Gross-Nachmani's Hyper Graph soft decoders based on Sourlas's Spin Glass reduction and Mezard's Replica Symmetry Breaking. Using it, we demonstrated reasons for uncertainty of the Neural Network loss function landscape and efficiency of replacing the arctanh neural network activation function with the Nishimori Temperature arctanh Taylor approximation. We compare the performance of short-length best known linear binary codes from Brouwer-Grassl codetable, Polar codes with sequence of frozen bits designed by Gaussian approximation and 5G eMBB Multi-Edge Type LDPC code with Base Graph 2 protograph on the AWGN-channel. The Sum-Product Flooding Scheduler decoder 50 iteration, Afterburn Saturated Min-Sum decoder, Ordered Statistics Decoder, Successive Cancellation Decoder with List size 32, Hyper Graph Neural Network under unfolding Belief Propagation decoder with Activation function Continues Metric Space relaxation according Nishimura temperature are used. The obtained simulation results are compared with the Finite-Length Polyanskiy theoretical boundary.