The Asymptotic Bit Error Probability of LDPC Codes for the Binary Erasure Channel with Finite Iteration Number

TL;DR

This paper derives an efficient algorithm to calculate the asymptotic difference in bit error probability for LDPC codes over the BEC at finite iterations, improving accuracy for small block lengths.

Contribution

It introduces a novel algorithm to compute the constant alpha, enhancing the understanding of finite-iteration decoding performance for regular LDPC ensembles.

Findings

The approximation with alpha is accurate even for small block lengths.

The algorithm efficiently computes alpha for regular ensembles.

Finite iteration error probability difference scales as alpha/n.

Abstract

We consider communication over the binary erasure channel (BEC) using low-density parity-check (LDPC) code and belief propagation (BP) decoding. The bit error probability for infinite block length is known by density evolution and it is well known that a difference between the bit error probability at finite iteration number for finite block length $n$ and for infinite block length is asymptotically $\alpha/n$, where $\alpha$ is a specific constant depending on the degree distribution, the iteration number and the erasure probability. Our main result is to derive an efficient algorithm for calculating $\alpha$ for regular ensembles. The approximation using $\alpha$ is accurate for $(2,r)$-regular ensembles even in small block length.