Analysis of Absorbing Sets and Fully Absorbing Sets of Array-Based LDPC Codes

• Published in: IEEE transactions on information theory Vol. 56; no. 1; pp. 181 - 201
• Main Authors: Dolecek, L., Zhengya Zhang, Anantharam, V., Wainwright, M.J., Nikolic, B.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.01.2010; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Absorbing set; Absorption; Algorithm design and analysis; Algorithms; Applied sciences; Bit error rate; bit-flipping; Blocking; Codes; Coding, codes; Combinatorial analysis; Decoding; error floor; Errors; Exact sciences and technology; Extrapolation; Flattening; Floors; Foundations; Information theory; Information, signal and communications theory; low-density parity-check (LDPC ) codes; Materials science and technology; Message passing; message passing decoding; near-codeword; Parity check codes; Performance analysis; Semiconductor materials; Signal and communications theory; Signal to noise ratio; Substructures; Telecommunications and information theory; trapping set; Bit flipping; Bit error rate; near-codeword; Theoretical study; error floor; Tanner graph; Decoding; Absorbing set; Combinatorial method; Trapping; message passing decoding; Flattening; Message passing; Algorithm performance; low-density parity-check (LDPC ) codes; Error correcting code; Parity check codes; trapping set; bit-flipping; Signal to noise ratio
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2009.2034781

The class of low-density parity-check (LDPC) codes is attractive, since such codes can be decoded using practical message-passing algorithms, and their performance is known to approach the Shannon limits for suitably large block lengths. For the intermediate block lengths relevant in applications, however, many LDPC codes exhibit a so-called ¿error floor,¿ corresponding to a significant flattening in the curve that relates signal-to-noise ratio (SNR) to the bit-error rate (BER) level. Previous work has linked this behavior to combinatorial substructures within the Tanner graph associated with an LDPC code, known as (fully) absorbing sets. These fully absorbing sets correspond to a particular type of near-codewords or trapping sets that are stable under bit-flipping operations, and exert the dominant effect on the low BER behavior of structured LDPC codes. This paper provides a detailed theoretical analysis of these (fully) absorbing sets for the class of Cp , ¿ array-based LDPC codes, including the characterization of all minimal (fully) absorbing sets for the array-based LDPC codes for ¿ = 2,3,4 , and moreover, it provides the development of techniques to enumerate them exactly. Theoretical results of this type provide a foundation for predicting and extrapolating the error floor behavior of LDPC codes.