Distance distributions in ensembles of irregular low-density parity-check codes

Abstract

We derive asymptotic expressions for the average distance distributions in ensembles of irregular low-density paritycheck (LDPC) codes. The ensembles are defined by matrices with given profiles of column and row sums. Index Terms-Distance distributions, low-density parity-check (LDPC) codes. I. INTRODUCTION T HE low-density parity-check (LDPC) codes are famous because of their performance in the vicinity of the Shannon limit under iterative decoding of modest complexity. However, this phenomenon is still far from being completely understood. One of the important parameters helping in analysis of codes' performance under maximum-likelihood decoding is their distance distribution. Such estimates are important in analyzing achievable limits of performance of LDPC codes, and optimization of their parameters. The problem of estimation distance distribution of regular codes (i.e., codes defined by parity-check matrices with fixed column and row sums) was addressed in many papers, starting with the original Gallager's work [2], see for most general results [5]. Use of such estimates to bounding performance of LDPC codes was discussed in [2], [7], [9], [13]. Recently, it was found that irregular LDPC codes (i.e., codes defined by parity-check matrices with several possible values of column and row sums) perform better than the regular codes under iterative decoding [6], [11], [12], [14]. In this paper, we solve the problem of estimation of the average distance distribution in ensembles of irregular LDPC codes. This question was addressed independently in [4], where an implicit expression for such distributions was given. Here we give an explicit formula describing the average distance distributions. The paper is organized as follows. We start by defining the considered ensemble of irregular codes, and define the average distance distribution in this ensemble. Then we count the number of matrices belonging to the ensemble of special shape, yielding that the defined codes contain the word having ones on the first positions and zeros on the remaining positions.

Asymptotic Spectra of Trapping Sets in Regular and Irregular LDPC Code Ensembles

“…The proof represents a generalization of the arguments used in [10] for evaluating average weight distributions of code ensembles. Again, we will work with transposes of parity-check matrices, H T .…”

“…, χ g , with the following set of properties. The first is the normalization property, m,n are identical to the ones considered in [9] and [10] for the purpose of evaluating the underlying codes average weight enumerators.…”

On Decoding of Low-Density Parity-Check Codes Over the Binary Erasure Channel

“…The distance distributions of irregular codes have been found by several authors [17], [18], and [14]. As shown in [16], if the minimum distance of the expurgated ensemble increases linearly with the code length.…”

“…Let be the time that Algorithm B needs to decode a received word and let be the average time of the decoding of the code using Algorithm B. We want to have (18) (19) where is a small constant close to zero and is a sufficiently small constant. Our simulations show that the algorithm we propose in this section (Algorithm C) will achieve the above inequalities with and .…”

Ensemble Weight Enumerators for Protograph-Based Generalized LDPC Codes

“…Because the conditions of the conjecture (same CN weight enumerators, same VN degree) hold, we will adopt the two observations above as assumptions. Let us define p (ρ) as the proportion of occurrence of a codeword of weight ρ in the constituent CN code, and letP = [p (0) , p (3) , p (4) , p (5) , p (6) , p (7) , p (8) , p (9) , p (10) , p (11) , p (12) , p (15) ]. By adopting the first observation above as an assumption, all elementsδ i inδ will have the same value, say δ ′ .…”