A Generalization of the Finite-Length Scaling Approach Beyond the BEC

We want to extend the approximation of the error probability via a scaling approach from the BEC to general binary-input memoryless output-symmetric (BMS) channels. In particular, we consider such scaling laws for regular LDPC ensembles and message-passing (MP) decoders with a finite number of messages. We first show how to re-derive the scaling law for transmission over the BEC using an “EXIT-like” curve instead of the density evolution curve of the peeling decoder. The advantage of the new derivation is that the new expression of the scaling parameter α only contains quantities that can be meaningfully interpreted also for general message-passing algorithms. In particular, this expression only depends on the curvature of the EXIT-like curve as well as the variance of the messages, both taken at the critical channel parameter. We discuss how to compute these quantities for general MP algorithms and we evaluate the expressions for the specific cases of the Gallager algorithm A as well as the Decoder with Erasures and compare the resulting predictions on the error probability with simulation results.