Doubly-Generalized LDPC Codes: Stability Bound over the BEC

TL;DR

This paper extends the stability bound for LDPC codes over the BEC to doubly-generalized LDPC codes with linear block component codes, establishing conditions for achieving the bound with equality.

Contribution

It develops a stability bound for doubly-generalized LDPC codes over the BEC, generalizing previous bounds to more complex node structures with maximum a posteriori erasure correction.

Findings

The stability bound depends only on component codes with minimum distance 2.

A derivative matching condition is identified for achieving the bound with equality.

The bound applies to a broad class of generalized LDPC codes.

Abstract

The iterative decoding threshold of low-density parity-check (LDPC) codes over the binary erasure channel (BEC) fulfills an upper bound depending only on the variable and check nodes with minimum distance 2. This bound is a consequence of the stability condition, and is here referred to as stability bound. In this paper, a stability bound over the BEC is developed for doubly-generalized LDPC codes, where the variable and the check nodes can be generic linear block codes, assuming maximum a posteriori erasure correction at each node. It is proved that in this generalized context as well the bound depends only on the variable and check component codes with minimum distance 2. A condition is also developed, namely the derivative matching condition, under which the bound is achieved with equality.