On Pseudocodewords and Improved Union Bound of Linear Programming Decoding of HDPC Codes

In this paper, we present an improved union bound on the Linear Programming (LP) decoding performance of the binary linear codes transmitted over an additive white Gaussian noise channels. The bounding technique is based on the second-order of Bonferroni-type inequality in probability theory, and it is minimized by Prim's minimum spanning tree algorithm. The bound calculation needs the fundamental cone generators of a given parity-check matrix rather than only their weight spectrum, but involves relatively low computational complexity. It is targeted to high-density parity-check codes, where the number of their generators is extremely large and these generators are spread densely in the Euclidean space. We explore the generator density and make a comparison between different parity-check matrix representations. That density effects on the improvement of the proposed bound over the conventional LP union bound. The paper also presents a complete pseudo-weight distribution of the fundamental cone generators for the BCH[31,21,5] code

LDPC Codes Which Can Correct Three Errors Under Iterative Decoding

In this paper, we provide necessary and sufficient conditions for a column-weight-three LDPC code to correct three errors when decoded using Gallager A algorithm. We then provide a construction technique which results in a code satisfying the above conditions. We also provide numerical assessment of code performance via simulation results.Comment: 5 pages, 3 figures, submitted to IEEE Information Theory Workshop (ITW), 200

Instanton-based Techniques for Analysis and Reduction of Error Floors of LDPC Codes

We describe a family of instanton-based optimization methods developed recently for the analysis of the error floors of low-density parity-check (LDPC) codes. Instantons are the most probable configurations of the channel noise which result in decoding failures. We show that the general idea and the respective optimization technique are applicable broadly to a variety of channels, discrete or continuous, and variety of sub-optimal decoders. Specifically, we consider: iterative belief propagation (BP) decoders, Gallager type decoders, and linear programming (LP) decoders performing over the additive white Gaussian noise channel (AWGNC) and the binary symmetric channel (BSC). The instanton analysis suggests that the underlying topological structures of the most probable instanton of the same code but different channels and decoders are related to each other. Armed with this understanding of the graphical structure of the instanton and its relation to the decoding failures, we suggest a method to construct codes whose Tanner graphs are free of these structures, and thus have less significant error floors.Comment: To appear in IEEE JSAC On Capacity Approaching Codes. 11 Pages and 6 Figure