Low-rate coding using incremental redundancy for GLDPC codes

Abstract

Abstract-In this paper we propose a low-rate coding method, suited for application-layer forward error correction. Depending on channel conditions, the coding scheme we propose can switch from a fixed-rate LDPC code to various low-rate GLDPC codes. The source symbols are first encoded by using a staircase or triangular LDPC code. If additional symbols are needed, the encoder is then switched to the GLDPC mode and extra-repair symbols are produced, on demand. In order to ensure small overheads, we consider irregular distributions of extra-repair symbols optimized by density evolution techniques. We also show that increasing the number of extra-repair symbols improves the successful decoding probability, which becomes very close to 1 for sufficiently many extra-repair symbols.

Optimization with exit functions of GLDPC-Staircase codes for the BEC

“…GLDPC-Staircase (N G , K) codes [9] [14] can be represented by a Tanner graph (Fig. 1) with the following meaning: that has the interesting property that the first repair symbol is also equal to the XOR sum of the source symbols.…”

“…The threshold value of a given ensemble of codes can be efficiently computed by using the Density Evolution (DE) method [10], which recursively computes the fraction of erased messages passed during the belief propagation decoding. Density evolution equations are derived in the next section, by using the methodology introduced in [9]. 1 In these tests the LDPC-Staircase code rate is set to r L = 2/3 and N 1 = 5.…”

“…Our work relies on the GLDPC-Staircase codes [9], a class of small rate GLDPC codes (i.e. that can efficiently produce a large number of repair symbols, on demand) with interesting erasure recovery performance under IT decoding.…”

“…(RS) scheme of [9], we add a Maximum Likelihood (ML) decoding scheme to further improve the code performance (the reasons why this is a good practical solution are detailed in Section 2.2). This extension, called hybrid decoding, results in exceptional performance gains (see Section 2.3).…”

Design of small rate, close to ideal, GLDPC-staircase AL-FEC codes for the erasure channel

“…Recently, a construction of GLDPC using LDPC Staircase code as base code and Reed-Solomon (RS) codes as component codes has been proposed in [12]. This construction allows each component code to produce a potentially large number of repair symbols in terms of RS codes (named extrarepair symbols) on demand, a feature that is well suited to situations where the channel conditions can be worse than expected, or to fountain like content distribution applications.…”