Finite-Length Scaling and Finite-Length Shift for Low-Density Parity-Check Codes

Consider communication over the binary erasure channel BEC(ǫ) using random low-density parity-check codes with finite-blocklength n from ‘standard’ ensembles. We show that large error events is conveniently described within a scaling theory, and explain how to estimate heuristically their effect. Among other quantities, we consider the finite length threshold ǫ∗(n), defined by requiring a block error probability PB = 1/2. For ensembles with minimum variable degree larger than two, the following expression is argued to hold ǫ(n) = ǫ − ǫ1 n +Θ(n) , with a calculable shift parameter ǫ1 > 0.