We describe the application of edge-local complementation (ELC) to a Tanner graph associated with a binary linear code, C. Various properties of ELC are described, mainly the special case of isomorphic ELC operations and the relationship to the automorphism group of the code, Aut(C), and the generalization of ELC to weight-bounding ELC (WB-ELC) operations under which the number of edges remains upper-bounded. ELC generates all systematic parity-check matrices (the orbit) of the code, so WB-ELC facilitates a restriction to low-weight matrices of this orbit. We propose using ELC and WB-ELC as a source of diversity, to improve iterative soft-input soft-output decoding of high-density parity-check (HDPC) codes, with the sum-product algorithm (SPA). A motivation of ELC-enhanced SPA decoding is locality; that diversity is achieved by local graph action, and is well-suited to the local actions that constitute the SPA and allows for parallel software implementation. Simulation data on the error-rate performance of the proposed SPA-ELC and SPA-WBELC iterative decoding algorithms is shown for several HDPC codes. A gain is reported over SPA decoding, and over a recently proposed algorithm to decode HDPC codes using permutations from Aut(C). ELC-enhanced decoding extends the scope of iterative decoding to codes with trivial Aut(C).

This work is an extension of our previous work on an iterative soft-decision decoder for high-density parity-check codes, using a graph-local operation known as edge-local complementation (ELC). Inferred least reliable codeword positions are targeted by an ELC stage in between sum-product algorithm iterations. A gain is shown over related iterative decoding algorithms - mainly due to an improved heuristic to determine optimum ELC locations in the Tanner graph - both in error-rate performance, as well as complexity in terms of a significant reduction in the required number of ELC operations. We also present a novel damping operation, generalized to the graph-local setting where extrinsic information remains on edges not affected by ELC.

In this paper, we propose to enhance the performance of the sum-product algorithm (SPA) by interleaving SPA iterations with a random local graph update rule. This rule is known as edge local complementation (ELC), and has the effect of modifying the Tanner graph while preserving the code. We have previously shown how the ELC operation can be used to implement an iterative permutation group decoder (SPA-PD)-one of the most successful iterative soft-decision decoding strategies at small blocklengths. In this work, we exploit the fact that ELC can also give structurally distinct parity-check matrices for the same code. Our aim is to describe a simple iterative decoder, running SPA-PD on distinct structures, based entirely on random usage of the ELC operation. This is called SPA-ELC, and we focus on small blocklength codes with strong algebraic structure. In particular, we look at the extended Golay code and two extended quadratic residue codes. Both error rate performance and average decoding complexity, measured by the average total number of messages required in the decoding, significantly outperform those of the standard SPA, and compares well with SPA-PD. However, in contrast to SPA-PD, which requires a global action on the Tanner graph, we obtain a performance improvement via local action alone. Such localized algorithms are of mathematical interest in their own right, but are also suited to parallel/distributed realizations.