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

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 $C_{p, \gamma}$ array-based LDPC codes, including the characterization of all minimal (fully) absorbing sets for the array-based LDPC codes for $\gamma = 2,3,4$, 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.