On the Tanner Graph Cycle Distribution of Random LDPC, Random Protograph-Based LDPC, and Random Quasi-Cyclic LDPC Code Ensembles

Abstract

In this paper, we study the cycle distribution of random low-density parity-check (LDPC) codes, randomly constructed protograph-based LDPC codes, and random quasi-cyclic (QC) LDPC codes. We prove that for a random bipartite graph, with a given (irregular) degree distribution, the distributions of cycles of different length tend to independent Poisson distributions, as the size of the graph tends to infinity. We derive asymptotic upper and lower bounds on the expected values of the Poisson distributions that are independent of the size of the graph, and only depend on the degree distribution and the cycle length. For a random lift of a bi-regular protograph, we prove that the asymptotic cycle distributions are essentially the same as those of random bipartite graphs as long as the degree distributions are identical. For random QC-LDPC codes, however, we show that the cycle distribution can be quite different from the other two categories. In particular, depending on the protograph and the value of c, the expected number of cycles of length c, in this case, can be either Θ(N ) or Θ(1), where N is the lifting degree (code length). We also provide numerical results that match our theoretical derivations.Our results provide a theoretical foundation for emperical results that were reported in the literature but were not well-justified. They can also be used for the analysis and design of LDPC codes and associated algorithms that are based on cycles.

Characterization and Efficient Search of Non-Elementary Trapping Sets of LDPC Codes With Applications to Stopping Sets

“…For S * to have the smallest size, we look for S i s with smallest a S i whose b S i values satisfy b S i > 0, and the constraint (2). It is easy to see that the most favorable candidate for TSs S i s is an ETSL with no cycle.…”

“…DISTANCE OR BOUNDS ON THE STOPPING DISTANCE OF SOME VARIABLE-REGULAR LDPC CODES Code dv Girth Rate Length Lower Bound Upper Bound b ′minutes to provide the lower and upper bounds of 7 and 9 on the s min of this code, respectively.Codes C 5 -C 8 are four high-rate random codes with variable degree 3 and girth 6 constructed by the program of [17] 2. These random high-rate codes with large block lengths are challenging codes for all the existing approaches in the literature.…”

Asymptotic Average Multiplicity of Structures Within Different Categories of Trapping Sets, Absorbing Sets, and Stopping Sets in Random Regular and Irregular LDPC Code Ensembles

“…In Table I, the value N 2a is used as an asymptotic approximation for the average number of TS structures that are isomorphic to a simple cycle of length 2a. The value N 2a was computed (approximated) recently in [24] for random regular (irregular) Tanner graphs.…”

Cospectral Bipartite Graphs with the Same Degree Sequences but with Different Number of Large Cycles

“…The connection between the performance of LDPC codes and cycles of the Tanner graph has motivated much research on the study of the cycle distribution and the counting of cycles in bipartite graphs, see, e.g., [2], [6], [7], [10], [19]. Counting cycles of a given length, even in bipartite graphs, is known to be NP-hard [24].…”

“…In [19], Karimi and Banihashemi presented an efficient message-passing algorithm to count the number of cycles of length less than 2g, in a general graph, where g is the girth of the graph. The distribution of cycles in different ensembles of bipartite graphs was studied in [6], where it was shown that for random ensembles of bipartite graphs, the multiplicities of cycles of different lengths have independent Poisson distributions with the expected values only a function of the cycle length and the degree distribution (and independent of the size of the graph). More recently, Blake and Lin [2] presented a formula to compute the multiplicity of cycles of length g in bi-regular bipartite graphs as a function of the spectrum (eigenvalues of the adjacency matrix of the graph) and degree sequences of the graph.…”