Fountain Uncorrectable Sets and Finite-Length Analysis

TL;DR

This paper introduces a novel analysis method for Fountain codes over the BEC by combining uncorrectable set and stopping set analysis, providing a more comprehensive understanding of decoding failure conditions.

Contribution

It proposes the concept of uncorrectable sets for Fountain codes and integrates it with stopping set analysis for finite-length performance evaluation.

Findings

Derived conditions for Fountain decoding failure over BEC

Analyzed uncorrectable sets in Fountain codes

Provided integrated performance analysis combining stopping and uncorrectable sets

Abstract

Decoding performance of Fountain codes for the binary erasure channel (BEC) depends on two aspects. One is the essential code structure, on which stopping set analysis operates. The other is the effect from the channel characteristic, which is difficult to give a precise estimation. To tackle these problems, in this paper, we propose a solution to analyzing the performance of Fountain codes based on the uncorrectable set. We give the condition for Fountain decoding failure over the BEC. Then, we conduct the analysis of uncorrectable set on Fountain codes. Finally, we combine the stopping set and the uncorrectable set to provide the integrated analysis on the performance of Fountain codes for BEC.