Mathematical Programming Decoding of Binary Linear Codes: Theory and Algorithms

Mathematical programming is a branch of applied mathematics and has recently been used to derive new decoding approaches, challenging established but often heuristic algorithms based on iterative message passing. Concepts from mathematical programming used in the context of decoding include linear, integer, and nonlinear programming, network flows, notions of duality as well as matroid and polyhedral theory. This survey article reviews and categorizes decoding methods based on mathematical programming approaches for binary linear codes over binary-input memoryless symmetric channels.Comment: 17 pages, submitted to the IEEE Transactions on Information Theory. Published July 201

Efficient Maximum-Likelihood Decoding of Linear Block Codes on Binary Memoryless Channels

In this work, we consider efficient maximum-likelihood decoding of linear block codes for small-to-moderate block lengths. The presented approach is a branch-and-bound algorithm using the cutting-plane approach of Zhang and Siegel (IEEE Trans. Inf. Theory, 2012) for obtaining lower bounds. We have compared our proposed algorithm to the state-of-the-art commercial integer program solver CPLEX, and for all considered codes our approach is faster for both low and high signal-to-noise ratios. For instance, for the benchmark (155,64) Tanner code our algorithm is more than 11 times as fast as CPLEX for an SNR of 1.0 dB on the additive white Gaussian noise channel. By a small modification, our algorithm can be used to calculate the minimum distance, which we have again verified to be much faster than using the CPLEX solver.Comment: Submitted to 2014 International Symposium on Information Theory. 5 Pages. Accepte

Minimum Pseudoweight Analysis of 3-Dimensional Turbo Codes

In this work, we consider pseudocodewords of (relaxed) linear programming (LP) decoding of 3-dimensional turbo codes (3D-TCs). We present a relaxed LP decoder for 3D-TCs, adapting the relaxed LP decoder for conventional turbo codes proposed by Feldman in his thesis. We show that the 3D-TC polytope is proper and $C$-symmetric, and make a connection to finite graph covers of the 3D-TC factor graph. This connection is used to show that the support set of any pseudocodeword is a stopping set of iterative decoding of 3D-TCs using maximum a posteriori constituent decoders on the binary erasure channel. Furthermore, we compute ensemble-average pseudoweight enumerators of 3D-TCs and perform a finite-length minimum pseudoweight analysis for small cover degrees. Also, an explicit description of the fundamental cone of the 3D-TC polytope is given. Finally, we present an extensive numerical study of small-to-medium block length 3D-TCs, which shows that 1) typically (i.e., in most cases) when the minimum distance $d_{\rm min}$ and/or the stopping distance $h_{\rm min}$ is high, the minimum pseudoweight (on the additive white Gaussian noise channel) is strictly smaller than both the $d_{\rm min}$ and the $h_{\rm min}$, and 2) the minimum pseudoweight grows with the block length, at least for small-to-medium block lengths.Comment: To appear in IEEE Transactions on Communication

Introduction to Mathematical Programming-Based Error-Correction Decoding

Decoding error-correctiong codes by methods of mathematical optimization, most importantly linear programming, has become an important alternative approach to both algebraic and iterative decoding methods since its introduction by Feldman et al. At first celebrated mainly for its analytical powers, real-world applications of LP decoding are now within reach thanks to most recent research. This document gives an elaborate introduction into both mathematical optimization and coding theory as well as a review of the contributions by which these two areas have found common ground.Comment: LaTeX sources maintained here: https://github.com/supermihi/lpdintr

Pseudocodewords of linear programming decoding of 3-dimensional turbo codes

In this work, we consider pseudocodewords of (relaxed) linear programming (LP) decoding of 3-dimensional turbo codes (3D-TCs), recently introduced by Berrou et al.. Here, we consider binary 3D-TCs while the original work of Berrou et al. considered double-binary codes. We present a relaxed LP decoder for 3D-TCs, which is an adaptation of the relaxed LP decoder for conventional turbo codes proposed by Feldman in his thesis. The vertices of this relaxed polytope are the pseudocodewords. We show that the support set of any pseudocodeword is a stopping set of iterative decoding of 3D-TCs using maximum a posteriori constituent decoders on the binary erasure channel. Furthermore, we present a numerical study of small block length 3D-TCs, which shows that typically the minimum pseudoweight (on the additive white Gaussian noise (AWGN) channel) is smaller than both the minimum distance and the stopping distance. In particular, we performed an exhaustive search over all interleaver pairs in the 3D-TC (with input block length K = 128) based on quadratic permutation polynomials over integer rings with a quadratic inverse. The search shows that the best minimum AWGN pseudoweight is strictly smaller than the best minimum/stopping distance

NMR Structural Profiling of Transcriptional Intermediates Reveals Riboswitch Regulation by Metastable RNA Conformations

Gene repression induced by the formation of transcriptional terminators represents a prime example for the coupling of RNA synthesis, folding, and regulation. In this context, mapping the changes in available conformational space of transcription intermediates during RNA synthesis is important to understand riboswitch function. A majority of riboswitches, an important class of small metabolite-sensing regulatory RNAs, act as transcriptional regulators, but the dependence of ligand binding and the subsequent allosteric conformational switch on mRNA transcript length has not yet been investigated. We show a strict fine-tuning of binding and sequence-dependent alterations of conformational space by structural analysis of all relevant transcription intermediates at single-nucleotide resolution for the I-A type 2′dG-sensing riboswitch from <i>Mesoplasma florum</i> by NMR spectroscopy. Our results provide a general framework to dissect the coupling of synthesis and folding essential for riboswitch function, revealing the importance of metastable states for RNA-based gene regulation

Noncovalent Spin Labeling of Riboswitch RNAs To Obtain Long-Range Structural NMR Restraints

Paramagnetic relaxation enhancement (PRE) NMR is a powerful method to study structure, dynamics and function of proteins. Up to now, the application of PRE NMR on RNAs is a significant challenge due to the limited size of chemically synthesized RNA. Here, we present a noncovalent spin labeling strategy to spin label RNAs in high yields required for NMR studies. The approach requires the presence of a helix segment composed of about 10 nucleotides (nt) but is not restricted by the size of the RNA. We show successful application of this strategy on the 2′dG sensing aptamer domain of <i>Mesoplasma florum</i> (78 nt). The aptamer domain was prepared in two fragments. A larger fragment was obtained by biochemical means, while a short spin labeled fragment was prepared by chemical solid-phase synthesis. The two fragments were annealed noncovalently by hybridization. We performed NMR, cw-EPR experiments and gel shift assays to investigate the stability of the two-fragment complex. NMR analysis in ¹⁵N-TROSY and ¹H,¹H-NOESY spectra of both unmodified and spin labeled aptamer domain show that the fragmented system forms a stable hybridization product, is in structural agreement with the full length aptamer domain and maintains its function. Together with structure modeling, experimentally determined ¹H-Γ₂ rates are in agreement with reported crystal structure data and show that distance restraints up to 25 Å can be obtained from NMR PRE data of RNA