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

摘要

Consider communication over the binary erasure channel BEC(ε) 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 ε~*(n), defined by requiring a block error probability P_B = 1/2. For ensembles with minimum variable degree larger than two, the following expression is argued to hold ε~* (n) = ε~* - ε_1~* n ~(-2/3) + Θ(n~(-1)), with a calculable shift parameter ε_1~* > 0.