Hardness Results on Finding Leafless Elementary Trapping Sets and Elementary Absorbing Sets of LDPC Codes

Abstract

Leafless elementary trapping sets (LETSs) are known to be the problematic structures in the error floor region of low-density parity-check (LDPC) codes over the additive white Gaussian (AWGN) channel under iterative decoding algorithms. While problems involving the general category of trapping sets, and the subcategory of elementary trapping sets (ETSs), have been shown to be NP-hard, similar results for LETSs, which are a subset of ETSs are not available. In this paper, we prove that, for a general LDPC code, finding a LETS of a given size a with minimum number of odd-degree check nodes b is NP-hard to approximate within any approximation factor. We also prove that finding the minimum size a of a LETS with a given b is NP-hard to approximate within any approximation factor. Similar results are proved for elementary absorbing sets, a popular subcategory of LETSs. Index Terms: Low-density parity-check (LDPC) codes, trapping sets (TS), elementary trapping sets (ETS), leafless elementary trapping sets (LETS), absorbing sets (ABS), elementary absorbing sets (EABS), computational complexity, NP-hardness.

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

“…Leafless ETSs (LETS) are known to be the most harmful TSs in the error floor region for variable-regular LDPC codes over the additive white Gaussian noise (AWGN) channel, see, [32] and the references therein. Finding LETSs, however, is in general a difficult problem [19]. Nevertheless, different characterizations of LETSs are provided in the literature [45], [32].…”