Random Edge-Local Complementation With Applications to Iterative Decoding of HDPC Codes

This paper describes the application of edge-local complementation (ELC), defined for a simple bipartite graph, to a Tanner graph associated with a binary linear code, C. From a code perspective, various properties of ELC are described and discussed, mainly the special case of isomorphic ELC operations and the relationship to the automorphism group of the code, Aut(C), as well as the generalization of ELC to weight-bounding ELC (WB-ELC) operations under which the number of edges remains upper-bounded. The main motivation is the use of ELC to improve iterative soft-input soft-output decoding of high-density parity-check (HDPC) codes using the sum-product algorithm (SPA). By updating the edges of the Tanner graph using ELC additional diversity is achieved, while maintaining control on the weight of the Tanner graph (which also influences the number of short cycles) via WB-ELC. One motivation of ELC-based SPA decoding is the locality argument; that diversity is achieved by local graph action, and so is well-suited to the local actions that constitute the SPA and allows a parallel implementation. Further applications of WB-ELC are described, including a heuristic to search for a systematic parity-check matrix (i.e., a Tanner graph) of reduced weight – a problem which has not received much focus in the literature. Extensive simulation data is shown for a range of HDPC codes, both in terms of matrix weight reduction, and error-rate performance of a proposed SPA-WBELC iterative decoding algorithm. A gain is reported over SPA decoding, and over a state-of-the-art algorithm to decode HDPC codes using permutations from Aut(C).