We analyze Linear Programming (LP) decoding of graphical binary codes operating over soft-output, symmetric and log-concave channels. We show that the error-surface, separating domain of the correct decoding from domain of the erroneous decoding, is a polytope. We formulate the problem of finding the lowest-weight pseudo-codeword as a non-convex optimization (maximization of a convex function) over a polytope, with the cost function defined by the channel and the polytope defined by the structure of the code. This formulation suggests new provably convergent heuristics for finding the lowest weight pseudo-codewords improving in quality upon previously discussed. The algorithm performance is tested on the example of the Tanner [155,64,20] code over the Additive White Gaussian Noise (AWGN) channel.

We consider linear programming (LP) decoding of a fixed low-density parity-check (LDPC) code over the binary symmetric channel (BSC). The LP decoder fails when it outputs a pseudo-codeword which is not a codeword. We propose an efficient algorithm termed the instanton search algorithm (ISA) which, given a random input, generates a set of flips called the BSC-instanton and prove that: (a) the LP decoder fails for any set of flips with support vector including an instanton; (b) for any input, the algorithm outputs an instanton in the number of steps upper-bounded by twice the number of flips in the input. We obtain the number of unique instantons of different sizes by running the ISA sufficient number of times. We then use the instanton statistics to predict the performance of the LP decoding over the BSC in the error floor region. We also propose an efficient semi-analytical method to predict the performance of LP decoding over a large range of transition probabilities of the BSC.

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.

In linear programming (LP) decoding of a low-density parity-check (LDPC) code one minimizes a linear functional, with coefficients related to log-likelihood ratios, over a relaxation of the polytope spanned by the codewords. In order to quantify LP decoding it is important to study vertexes of the relaxed polytope, so-called pseudocodewords. We propose a technique to heuristcally create a list of pseudocodewords close to the zero codeword and their distances. Our pseudocodeword-search algorithm starts by randomly choosing configuration of the noise. The configuration is modified through a discrete number of steps. Each step consists of two substeps: one applies an LP decoder to the noise-configuration deriving a pseudocodeword, and then finds configuration of the noise equidistant from the pseudocodeword and the zero codeword. The resulting noise configuration is used as an entry for the next step. The iterations converge rapidly to a pseudocodeword neighboring the zero codeword. Repeated many times, this procedure is characterized by the distribution function of the pseudocodeword effective distance. The efficiency of the procedure is demonstrated on examples of the Tanner code and Margulis codes operating over an additive white Gaussian noise (AWGN) channel.

We discuss the performance of low-density-parity-check (LDPC) codes decoded by means of linear programming (LP) at moderate and large signal-to-noise-ratios (SNR). Utilizing a combination of the previously introduced pseudo-codeword-search method and a new "dendro" trick, which allows us to reduce the complexity of the LP decoding, we analyze the dependence of the frame-error-rate (FER) on the SNR. Under maximum-a-posteriori (MAP) decoding the dendro-code, having only checks with connectivity degree three, performs identically to its original code with high-connectivity checks. For a number of popular LDPC codes performing over the additive-white-Gaussian-noise (AWGN) channel we found that either an error-floor sets at a relatively low SNR, or otherwise a transient asymptote, characterized by a faster decay of FER with the SNR increase, precedes the error-floor asymptote. We explain these regimes in terms of the pseudo-codeword spectra of the codes.

In this paper we develop instanton method introduced in V. Chernyak et al. (2004), M.G. Stepanov et al. (2005) to analyze quantitatively performance of low-density parity-check (LDPC) codes decoded iteratively in the so-called error-floor regime. We discuss statistical properties of the numerical instanton-amoeba scheme focusing on detailed analysis and comparison of two regular LDPC codes: Tanner's [155,64,20] and Margulis' [672,336,16] codes. In the regime of moderate values of the signal-to-noise ratio we critically compare results of the instanton-amoeba evaluations against the standard Monte Carlo calculations of the frame-error-rate

We present a theoretical tool for evaluation of error code performance on graphs. The method is known under the name of instanton calculus and is common in theoretical physics. We introduce the instanton calculus for linear block codes, and give a closed form expression for the bit error rate for a class of codes whose graphical model is approximated locally by a tree.