Efficient Algorithms to Find All Small Error-Prone Substructures in LDPC Codes

Abstract

Exhaustively enumerating all small error-prone sub structures in arbitrary, finite-length low-density parity-check (LDPC) codes has been proven to be NP-complete. In this paper, we present two exhaustive search algorithms to find such small error-prone substructures of an arbitrary LDPC code given its parity-check matrix. One algorithm is guaranteed to find all error-prone substructures including stopping sets, trapping sets, and absorbing sets, which have no more than α <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">max</sub> variable nodes and up to b <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">max</sub> induced odd-degree neighboring check nodes. The other algorithm is specially designed to find fully absorbing sets (FAS). Numerical results show that both of our proposed algorithms are more efficient in terms of execution time than another recently proposed exhaustive search algorithm [13]. Moreover, by properly initialization of the algorithm, the efficiency can be further improved for quasi-cyclic (QC) codes.