Generating parity check equations for bounded-distance iterative erasure decoding

TL;DR

This paper introduces a method to generate parity check equations for bounded-distance iterative erasure decoding, providing explicit constructions and analyzing the minimal size of such sets to improve decoding performance.

Contribution

It presents an explicit construction of generic (r,m)-erasure correcting sets and proves their minimal size grows linearly with r for fixed m.

Findings

Explicit construction of (r,m)-erasure correcting sets

Minimal size of sets is linear in r for fixed m

Analysis of stopping sets in iterative decoding

Abstract

A generic $(r,m)$-erasure correcting set is a collection of vectors in $\bF_2^r$ which can be used to generate, for each binary linear code of codimension $r$, a collection of parity check equations that enables iterative decoding of all correctable erasure patterns of size at most $m$. That is to say, the only stopping sets of size at most $m$ for the generated parity check equations are the erasure patterns for which there is more than one manner to fill in theerasures to obtain a codeword. We give an explicit construction of generic $(r,m)$-erasure correcting sets of cardinality $\sum_{i=0}^{m-1} {r-1\choose i}$. Using a random-coding-like argument, we show that for fixed $m$, the minimum size of a generic $(r,m)$-erasure correcting set is linear in $r$. Keywords: iterative decoding, binary erasure channel, stopping set