Efficient Search and Elimination of Harmful Objects for the Optimization of QC-SC-LDPC Codes

Abstract

The error correction performance of low-density parity-check (LDPC) codes under iterative message-passing decoding is degraded by the presence of certain harmful objects existing in their Tanner graph representation. Depending on the context, such harmful objects are known as stopping sets, trapping sets, absorbing sets, or pseudocodewords. In this paper, we propose a general procedure based on edge spreading that enables the design of quasi-cyclic (QC) spatially coupled lowdensity parity-check codes (SC-LDPCCs) that are derived from QC-LDPC block codes and possess a significantly reduced multiplicity of harmful objects with respect to the original QC-LDPC block code. This procedure relies on a novel algorithm that greedily spans the search space of potential candidates to reduce the multiplicity of the target harmful object(s) in the Tanner graph.The effectiveness of the method we propose is validated via examples and numerical computer simulations.

Error Floor Analysis of Quasi-Cyclic LDPC and Spatially Coupled-LDPC Codes and Construction of Codes with Low Error Floor

“…In order to identify if S is twice-LSS expandable, one can Proof. Comparing the range R 3 with the smaller range R 2 , one can verify that R 3 includes three extra classes of LETSs: (7, 3), (8, 0), and (8,2). Since the constraints in this proposition satisfy all the constraints of Proposition 6, we know that all the LETSs within R 2 are removed.…”

“…In Fig. 5.10, we have also shown the parent structures in (3,6), (4,8) and (5,6) classes outside the range. From the parent/child relationships, it is clear that if the (4, 4) structure is eliminated and the simple 8-cycles are dot m -nE for m = 3 and m = 4, then all the structures in R 4 will be eliminated.…”

“…Finite length construction of SC-LDPC codes has been also investigated in several research works including [1], [24], [62], [10], [26], [8], [30], where the authors moved beyond the girth and considered other error-prone structures as well, to design highperformance finite-length SC codes. However, in all of these works, the problem is addressed indirectly through a minimization over the small cycles contributing to those harmful TSs.…”

“…In this regard, the authors developed a design approach with the aim of minimizing the multiplicity of 8-cycles, as they demonstrated that 8-cycles are considered as the common denominator of most harmful structures for CB SC codes over PR channels. Most recently, the work in [8] has been devoted to the construction of QC SC-LDPC codes for a given underlying QC-LDPC block code such that the multiplicity of cycles of specific lengths, contributing to harmful structures in the underlying block code, are minimized. Schmalen et al in [76] introduced a number of design options to construct SC-LDPC codes and assessed how fast they are decodable by optimizing the contributing parameters.…”

“…First of all, we consider different values of γ and c and calculate the multiplicity of cycles (i.e., TBC walks) of length 4 and 6 in these fully-connected γ × c base graphs. 8 This value for a cycle of length k, which is called as N block base graph k , could be found using [18,Theorem 5] or through a computer search. Then, we generate a number of γ × c spreading matrices whose entries are assigned independently and identically with a uniform distribution in [0, m].…”