Computing the Asymptotic Expected Multiplicity of Elementary Trapping Sets (ETSs) in Random LDPC Code Ensembles

Abstract

The performance of low-density parity-check (LDPC) codes in the error floor region is closely related to some substructures of the code’s Tanner graple (simple) cycle have a nonzero finite average multiplicity. In this paper, we compute the asymptotic expected multiplicity of such ETS structures in random LDPC code ensembles. The computation, in general, involves two subproblems: (i) counting the number of cycles of a certain length with different combinations of variable node degrees, and (ii) counting the number of ways trees of different sizes can be appended to such a cycle to form an ETS. The first subproblem, which counts the number of leafless ETSs (LETSs), is solved by the authors in other papers. The second subproblem is formulated as a recursive counting problem and is solved in this paper, with the solution being a generalization of Catalan numbers. We also demonstrate through numerical results that the asymptotic expected values computed in this paper match the multiplicity of ETSs in randomly selected finite-length LDPC codes, even at relatively short block lengths.