Analysis of Absorbing Sets and Fully Absorbing Sets of Array-Based LDPC Codes

Abstract

The class of low-density parity-check (LDPC) codes is attractive, since such codes can be decoded using practical message-passing algorithms, and their performance is known to approach the Shannon limits for suitably large block lengths. For the intermediate block lengths relevant in applications, however, many LDPC codes exhibit a so-called "error floor," corresponding to a significant flattening in the curve that relates signal-to-noise ratio (SNR) to the bit-error rate (BER) level. Previous work has linked this behavior to combinatorial substructures within the Tanner graph associated with an LDPC code, known as (fully) absorbing sets. These fully absorbing sets correspond to a particular type of near-codewords or trapping sets that are stable under bit-flipping operations, and exert the dominant effect on the low BER behavior of structured LDPC codes. This paper provides a detailed theoretical analysis of these (fully) absorbing sets for the class of Cp; array-based LDPC codes, including the characterization of all minimal (fully) absorbing sets for the array-based LDPC codes for = 2; 3; 4, and moreover, it provides the development of techniques to enumerate them exactly. Theoretical results of this type provide a foundation for predicting and extrapolating the error floor behavior of LDPC codes. Index Terms-Absorbing set, bit-flipping, error floor, low-density parity-check (LDPC) codes, message passing decoding, nearcodeword, trapping set. I. INTRODUCTION L OW-density parity-check (LDPC) codes are a class of error-correcting codes based on sparse graphs. Their chief appeal is their excellent performance under practical decoding algorithms based on message passing, especially for

Towards improved LDPC code designs using absorbing set spectrum properties

“…We first show in Lemma 1 that (4, 8) absorbing sets indeed exist for this r = 5 array-based code family. This theoretical result is also consistent with previous experimental results of a sum-product decoding algorithm built in software and on a hardware emulator [30] for which it was shown that decoding errors due to (4,8) absorbing sets dominate the low BER region. This error floor region is determined by noncodewords (absorbing sets) whose weight is strictly smaller than the minimum distance of the code [27].…”

“…Previous work [4], [20] studied structural properties of dominant absorbing sets for a representative class of practical regular LDPC codes. Concurrently, absorbing sets were also experimentally verified on a hardware emulator [28], [30] to govern the low BER region performance for several representative LDPC code families.…”

“…This section reviews absorbing sets, [4], [28] and describes a class of high-performance LDPC codes that serves as our illustrative case study.…”

Finite alphabet iterative decoders approaching maximum likelihood performance on the Binary Symmetric Channel

“…Typically it has been observed that, in the case of regular d v = 3 LDPC codes, most harmful (a, b) trapping sets have small values of b. Note that this is necessarily the case for LDPC codes with d v = 4, as explained with the concept of absorbing sets in [18]. In addition, the value of A should be chosen depending on the value of guaranteed error correction capability t that we are trying to achieve, and based on the following conjecture.…”

Trapping set structure of LDPC codes on finite geometries

“…Let G be the Tanner graph of a binary LDPC code C given by the null space of an m × n matrix H over GF (2). For 1 ≤ κ ≤ n and 0 ≤ τ ≤ m, we have following definitions [3], [8], [11] [11].…”