Doubly-Generalized LDPC Codes: Stability Bound Over the BEC

• Published in: IEEE transactions on information theory Vol. 55; no. 3; pp. 1027 - 1046
• Main Authors: Paolini, E., Fossorier, M., Chiani, M.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.03.2009; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Applied sciences; Binary erasure channel (BEC); Bipartite graph; Block codes; Channels; Codes; Coding, codes; Computer Science; Cryptography; Data encryption; Decoding; Derivatives; Engineering Sciences; Equations; Error correction codes; error-correcting codes; Exact sciences and technology; extrinsic information transfer (EXIT) chart; information function; Information theory; Information, signal and communications theory; Iterative decoding; Iterative methods; low-density parity-check (LDPC) codes; Parity check codes; Signal and communications theory; Signal and Image Processing; Stability; stability condition; Systems, networks and services of telecommunications; Telecommunications; Telecommunications and information theory; Thresholds; Transmission and modulation (techniques and equipments); Upper bound; Upper bounds; Block code; Binary channel; Code distance; Stability criterion; extrinsic information transfer (EXIT) chart; Information transmission; Iterative decoding; stability condition; Linear code; error-correcting codes; Upper bound; Binary erasure channel (BEC); low-density parity-check (LDPC) codes; information function; A posteriori estimation; Parity check codes; Error correcting code; Minimal distance
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2008.2011446

The iterative decoding threshold of low-density parity-check (LDPC) codes over the binary erasure channel (BEC) fulfills an upper bound depending only on the variable and check nodes with minimum distance 2. This bound is a consequence of the stability condition, and is here referred to as stability bound. In this paper, a stability bound over the BEC is developed for doubly-generalized LDPC codes, where variable and check nodes can be generic linear block codes, assuming maximum a posteriori erasure correction at each node. It is proved that also in this generalized context the bound depends only on the variable and check component codes with minimum distance 2. A condition is also developed, namely, the derivative matching condition, under which the bound is achieved with equality. The stability bound leads to consider single parity-check codes used as variable nodes as an appealing option to overcome common problems created by generalized check nodes.