Ensemble Trapping Set Enumerators for Protograph-Based LDPC Codes

Protograph-based LDPC codes have the advantages of a simple design (or search) procedure and highly structured encoders and decoders. Recently, techniques fo r finding asymptotic trapping set enumerators in regular and irregular LDPC code ensembles was published [1]. In this pap er, finite-length ensemble trapping set enumerators for protographbased LDPC codes are obtained. The asymptotic results are derived from the finite-length results by letting the block length go to infinity. The asymptotic results allow us to determine whether or not the typical smallest trapping set size grows linearly with codeword length. I. I NTRODUCTION Graphical configurations associated with so-called Trapping sets in the Tanner graph of an LDPC code are known to create an error-rate floor in the performance curves for iterative decoders [1]–[5]. In [2], Richardson establishe d the notion of trapping sets and suggested a two-phase technique for predicting the performance of LDPC codes in the errorfloor region. The first phase searches for trapping sets and their multiplicities (e.g., by simulation) and the second p hase evaluate their contribution to the error floor. The authors o f [3], [4] used a similar technique to predict the floor of sever al LDPC codes on the binary symmetric channels. In [1], the authors derive the average asymptotic behavior of trapping set enumerators over random, regular and irregular, LDPC code ensembles. In this paper, we leverage our earlier work for evaluating weight and stopping set enumerators [6]–[8] to compute the trapping set enumerators for protograph-based LPDC codes. We consider both finite-length ensembles and infinite-lengt h ensembles. Together with Richardson’s performance evalua tion technique, these results allow one to estimate the aver age performance of LDPC codes in an ensemble characterized by a protograph. Moreover, simulation results in [2]–[5] show ed that small trapping sets dominates the error-floor level. In this work, we show how to use the asymptotic enumerator results to determine whether or not the typical smallest trapping set size grows linearly with codeword length, i.e. , whether or not the ensemble avoids small trapping sets for fairly long codes. Following [1], we derive bothgeneral and elementarytrapping set enumerators, but [1] does not consider protograph-based LDPC code ensembles. We define general and elementary trapping sets in Sections III and IV, respectively. Fig. 1. Vectorized protograph. The paper is organized as follows: In the next section a review of the method of finding weight enumerators for GLDPC code ensembles is presented. In Section III, general trapping set enumerators is evaluated, and in Section IV enumerators are found for the elementary trapping set case. Numerical results are presented in Section V. Finally, the paper is concluded in Section VI. II. W EIGHT ENUMERATORS FORPROTOGRAPH-BASED LDPC CODE ENSEMBLES The weight enumerators formulas in [7], [8] are used in this paper to compute trapping set enumerators. Consequently, in this section we present a review of our method for finding finite and asymptotic weight enumerators for protograph-based LDPC code ensembles. (As in [7], [8], the results in this section apply to generalized LDPC codes.) Recall that a protograph [9], [10] is a relatively small bipartite graph, containing variable nodes (VNs) and constra int nodes (CNs), from which a larger graph can be obtained by a copy-and-permute procedure: the protograph is copied N times, and then the edges of the individual replicas are permuted among the replicas to obtain a single, large graph. The copy-and-permute process can be simply represented by replacing each node with a vector of nodes of the same type a d replacing each edge with a bundle of (permuted) edges of the same type. This “vectorized” protograph is depicted in Fig. 1. A. Finite-length ensemble weight enumerators Following the notation of [6]–[8], consider an LDPC protograph,G = (V,C,E), whereV = {v1,v2,. , vnv} is the set ofnv VNs, C = {c1,c2,. ,cnc} is the set of nc CNs, andE is the set of edges. Denote by qvi (qcj ) the degree of variable (constraint) node vi (cj). Now consider the LDPC code constructed from a protograph G by makingN replicas ofG and using uniform interleavers, each of size N , to permute the edges among the replicas of the protograph. In order to exploit the results in [11], [12], we treat the VNs and CNs as constituent codes in a concatenated coding scheme. More specifically, the group of N VNs of typevi is considered to be a constituent (repetition) code with a weig htdi input of lengthN and qvi length-N outputs. Also, the group ofN CNs of typecj is considered to be a constituent code withqcj inputs, each of length N , and a fictitious output of weight zero. Let A(d) be the average (over the ensemble) number of codewords having weight vector d = [d1, d2, . , dnv ] corresponding to thenv VN N -groups and satisfying the protograph constraints. A(d) is thevector weight enumerator for the ensemble of codes of length N · nv described by the protograph. Let us further define