Iterative Decoding on Multiple Tanner Graphs Using Random Edge Local Complementation

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.Comment: 5 pages, to appear in proc. IEEE ISIT, June 200

Analysis of the Min-Sum Algorithm for Packing and Covering Problems via Linear Programming

Message-passing algorithms based on belief-propagation (BP) are successfully used in many applications including decoding error correcting codes and solving constraint satisfaction and inference problems. BP-based algorithms operate over graph representations, called factor graphs, that are used to model the input. Although in many cases BP-based algorithms exhibit impressive empirical results, not much has been proved when the factor graphs have cycles. This work deals with packing and covering integer programs in which the constraint matrix is zero-one, the constraint vector is integral, and the variables are subject to box constraints. We study the performance of the min-sum algorithm when applied to the corresponding factor graph models of packing and covering LPs. We compare the solutions computed by the min-sum algorithm for packing and covering problems to the optimal solutions of the corresponding linear programming (LP) relaxations. In particular, we prove that if the LP has an optimal fractional solution, then for each fractional component, the min-sum algorithm either computes multiple solutions or the solution oscillates below and above the fraction. This implies that the min-sum algorithm computes the optimal integral solution only if the LP has a unique optimal solution that is integral. The converse is not true in general. For a special case of packing and covering problems, we prove that if the LP has a unique optimal solution that is integral and on the boundary of the box constraints, then the min-sum algorithm computes the optimal solution in pseudo-polynomial time. Our results unify and extend recent results for the maximum weight matching problem by [Sanghavi et al.,'2011] and [Bayati et al., 2011] and for the maximum weight independent set problem [Sanghavi et al.'2009]

Efficient soft decoding techniques for reed-solomon codes

The main focus of this thesis is on finding efficient decoding methods for Reed-Solomon (RS) codes, i.e., algorithms with acceptable performance and affordable complexity. Three classes of decoders are considered including sphere decoding, belief propagation decoding and interpolation-based decoding. Originally proposed for finding the exact solution of least-squares problems, sphere decoding (SD) is used along with the most reliable basis (MRB) to design an efficient soft decoding algorithm for RS codes. For an (N, K ) RS code, given the received vector and the lattice of all possible transmitted vectors, we propose to look for only those lattice points that fall within a sphere centered at the received vector and also are valid codewords. To achieve this goal, we use the fact that RS codes are maximum distance separable (MDS). Therefore, we use sphere decoding in order to find tentative solutions consisting of the K most reliable code symbols that fall inside the sphere. The acceptable values for each of these symbols are selected from an ordered set of most probable transmitted symbols. Based on the MDS property, K code symbols of each tentative solution can he used to find the rest of codeword symbols. If the resulting codeword is within the search radius, it is saved as a candidate transmitted codeword. Since we first find the most reliable code symbols and for each of them we use an ordered set of most probable transmitted symbols, candidate codewords are found quickly resulting in reduced complexity. Considerable coding gains are achieved over the traditional hard decision decoders with moderate increase in complexity. Due to their simplicity and good performance when used for decoding low density parity check (LDPC) codes, iterative decoders based on belief propagation (BP) have also been considered for RS codes. However, the parity check matrix of RS codes is very dense resulting in lots of short cycles in the factor graph and consequently preventing the reliability updates (using BP) from converging to a codeword. In this thesis, we propose two BP based decoding methods. In both of them, a low density extended parity check matrix is used because of its lower number of short cycles. In the first method, the cyclic structure of RS codes is taken into account and BP algorithm is applied on different cyclically shifted versions of received reliabilities, capable of detecting different error patterns. This way, some deterministic errors can be avoided. The second method is based on information correction in BP decoding where all possible values are tested for selected bits with low reliabilities. This way, the chance of BP iterations to converge to a codeword is improved significantly. Compared to the existing iterative methods for RS codes, our proposed methods provide a very good trade-off between the performance and the complexity. We also consider interpolation based decoding of RS codes. We specifically focus on Guruswami-Sudan (GS) interpolation decoding algorithm. Using the algebraic structure of RS codes and bivariate interpolation, the GS method has shown improved error correction capability compared to the traditional hard decision decoders. Based on the GS method, a multivariate interpolation decoding method is proposed for decoding interleaved RS (IRS) codes. Using this method all the RS codewords of the interleaved scheme are decoded simultaneously. In the presence of burst errors, the proposed method has improved correction capability compared to the GS method. This method is applied for decoding IRS codes when used as outer codes in concatenated code

Bit flipping decoding for binary product codes

Error control coding has been used to mitigate the impact of noise on the wireless channel. Today, wireless communication systems have in their design Forward Error Correction (FEC) techniques to help reduce the amount of retransmitted data. When designing a coding scheme, three challenges need to be addressed, the error correcting capability of the code, the decoding complexity of the code and the delay introduced by the coding scheme. While it is easy to design coding schemes with a large error correcting capability, it is a challenge finding decoding algorithms for these coding schemes. Generally increasing the length of a block code increases its error correcting capability and its decoding complexity. Product codes have been identified as a means to increase the block length of simpler codes, yet keep their decoding complexity low. Bit flipping decoding has been identified as simple to implement decoding algorithm. Research has generally been focused on improving bit flipping decoding for Low Density Parity Check codes. In this study we develop a new decoding algorithm based on syndrome checking and bit flipping to use for binary product codes, to address the major challenge of coding systems, i.e., developing codes with a large error correcting capability yet have a low decoding complexity. Simulated results show that the proposed decoding algorithm outperforms the conventional decoding algorithm proposed by P. Elias in BER and more significantly in WER performance. The algorithm offers comparable complexity to the conventional algorithm in the Rayleigh fading channel

Unreliable and resource-constrained decoding

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2010.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student submitted PDF version of thesis.Includes bibliographical references (p. 185-213).Traditional information theory and communication theory assume that decoders are noiseless and operate without transient or permanent faults. Decoders are also traditionally assumed to be unconstrained in physical resources like material, memory, and energy. This thesis studies how constraining reliability and resources in the decoder limits the performance of communication systems. Five communication problems are investigated. Broadly speaking these are communication using decoders that are wiring cost-limited, that are memory-limited, that are noisy, that fail catastrophically, and that simultaneously harvest information and energy. For each of these problems, fundamental trade-offs between communication system performance and reliability or resource consumption are established. For decoding repetition codes using consensus decoding circuits, the optimal tradeoff between decoding speed and quadratic wiring cost is defined and established. Designing optimal circuits is shown to be NP-complete, but is carried out for small circuit size. The natural relaxation to the integer circuit design problem is shown to be a reverse convex program. Random circuit topologies are also investigated. Uncoded transmission is investigated when a population of heterogeneous sources must be categorized due to decoder memory constraints. Quantizers that are optimal for mean Bayes risk error, a novel fidelity criterion, are designed. Human decision making in segregated populations is also studied with this framework. The ratio between the costs of false alarms and missed detections is also shown to fundamentally affect the essential nature of discrimination. The effect of noise on iterative message-passing decoders for low-density parity check (LDPC) codes is studied. Concentration of decoding performance around its average is shown to hold. Density evolution equations for noisy decoders are derived. Decoding thresholds degrade smoothly as decoder noise increases, and in certain cases, arbitrarily small final error probability is achievable despite decoder noisiness. Precise information storage capacity results for reliable memory systems constructed from unreliable components are also provided. Limits to communicating over systems that fail at random times are established. Communication with arbitrarily small probability of error is not possible, but schemes that optimize transmission volume communicated at fixed maximum message error probabilities are determined. System state feedback is shown not to improve performance. For optimal communication with decoders that simultaneously harvest information and energy, a coding theorem that establishes the fundamental trade-off between the rates at which energy and reliable information can be transmitted over a single line is proven. The capacity-power function is computed for several channels; it is non-increasing and concave.by Lav R. Varshney.Ph.D