Exact Free Distance and Trapping Set Growth Rates for LDPC Convolutional Codes

Ensembles of (J,K)-regular low-density parity-check convolutional (LDPCC) codes are known to be asymptotically good, in the sense that the minimum free distance grows linearly with the constraint length. In this paper, we use a protograph-based analysis of terminated LDPCC codes to obtain an upper bound on the free distance growth rate of ensembles of periodically time-varying LDPCC codes. This bound is compared to a lower bound and evaluated numerically. It is found that, for a sufficiently large period, the bounds coincide. This approach is then extended to obtain bounds on the trapping set numbers, which define the size of the smallest, non-empty trapping sets, for these asymptotically good, periodically time-varying LDPCC code ensembles.Comment: To be presented at the 2011 IEEE International Symposium on Information Theor

On the Minimum Distance of Generalized Spatially Coupled LDPC Codes

Families of generalized spatially-coupled low-density parity-check (GSC-LDPC) code ensembles can be formed by terminating protograph-based generalized LDPC convolutional (GLDPCC) codes. It has previously been shown that ensembles of GSC-LDPC codes constructed from a protograph have better iterative decoding thresholds than their block code counterparts, and that, for large termination lengths, their thresholds coincide with the maximum a-posteriori (MAP) decoding threshold of the underlying generalized LDPC block code ensemble. Here we show that, in addition to their excellent iterative decoding thresholds, ensembles of GSC-LDPC codes are asymptotically good and have large minimum distance growth rates.Comment: Submitted to the IEEE International Symposium on Information Theory 201

Spatially Coupled LDPC Codes Constructed from Protographs

In this paper, we construct protograph-based spatially coupled low-density parity-check (SC-LDPC) codes by coupling together a series of L disjoint, or uncoupled, LDPC code Tanner graphs into a single coupled chain. By varying L, we obtain a flexible family of code ensembles with varying rates and frame lengths that can share the same encoding and decoding architecture for arbitrary L. We demonstrate that the resulting codes combine the best features of optimized irregular and regular codes in one design: capacity approaching iterative belief propagation (BP) decoding thresholds and linear growth of minimum distance with block length. In particular, we show that, for sufficiently large L, the BP thresholds on both the binary erasure channel (BEC) and the binary-input additive white Gaussian noise channel (AWGNC) saturate to a particular value significantly better than the BP decoding threshold and numerically indistinguishable from the optimal maximum a-posteriori (MAP) decoding threshold of the uncoupled LDPC code. When all variable nodes in the coupled chain have degree greater than two, asymptotically the error probability converges at least doubly exponentially with decoding iterations and we obtain sequences of asymptotically good LDPC codes with fast convergence rates and BP thresholds close to the Shannon limit. Further, the gap to capacity decreases as the density of the graph increases, opening up a new way to construct capacity achieving codes on memoryless binary-input symmetric-output (MBS) channels with low-complexity BP decoding.Comment: Submitted to the IEEE Transactions on Information Theor

Asymptotically Good LDPC Convolutional Codes Based on Protographs

LDPC convolutional codes have been shown to be capable of achieving the same capacity-approaching performance as LDPC block codes with iterative message-passing decoding. In this paper, asymptotic methods are used to calculate a lower bound on the free distance for several ensembles of asymptotically good protograph-based LDPC convolutional codes. Further, we show that the free distance to constraint length ratio of the LDPC convolutional codes exceeds the minimum distance to block length ratio of corresponding LDPC block codes.Comment: Proceedings of the 2008 IEEE International Symposium on Information Theory, Toronto, ON, Canada, July 6 - 11, 200

Spatially coupled generalized LDPC codes: asymptotic analysis and finite length scaling

Generalized low-density parity-check (GLDPC) codes are a class of LDPC codes in which the standard single parity check (SPC) constraints are replaced by constraints defined by a linear block code. These stronger constraints typically result in improved error floor performance, due to better minimum distance and trapping set properties, at a cost of some increased decoding complexity. In this paper, we study spatially coupled generalized low-density parity-check (SC-GLDPC) codes and present a comprehensive analysis of these codes, including: (1) an iterative decoding threshold analysis of SC-GLDPC code ensembles demonstrating capacity approaching thresholds via the threshold saturation effect; (2) an asymptotic analysis of the minimum distance and free distance properties of SC-GLDPC code ensembles, demonstrating that the ensembles are asymptotically good; and (3) an analysis of the finite-length scaling behavior of both GLDPC block codes and SC-GLDPC codes based on a peeling decoder (PD) operating on a binary erasure channel (BEC). Results are compared to GLDPC block codes, and the advantages and disadvantages of SC-GLDPC codes are discussed.This work was supported in part by the National Science Foundation under Grant ECCS-1710920, Grant OIA-1757207, and Grant HRD-1914635; in part by the European Research Council (ERC) through the European Union's Horizon 2020 research and innovation program under Grant 714161; and in part by the Spanish Ministry of Science, Innovation and University under Grant TEC2016-78434-C3-3-R (AEI/FEDER, EU)

Mathematical approach to channel codes with a diagonal matrix structure

Digital communications have now become a fundamental part of modern society. In communications, channel coding is an effective way to reduce the information rate down to channel capacity so that the information can be transmitted reliably through the channel. This thesis is devoted to studying the mathematical theory and analysis of channel codes that possess a useful diagonal structure in the parity-check and generator matrices. The first aspect of these codes that is studied is the ability to describe the parity-check matrix of a code with sliding diagonal structure using polynomials. Using this framework, an efficient new method is proposed to obtain a generator matrix G from certain types of parity-check matrices with a so-called defective cyclic block structure. By the nature of this method, G can also be completely described by a polynomial, which leads to efficient encoder design using shift registers. In addition, there is no need for the matrices to be in systematic form, thus avoiding the need for Gaussian elimination. Following this work, we proceed to explore some of the properties of diagonally structured lowdensity parity-check (LDPC) convolutional codes. LDPC convolutional codes have been shown to be capable of achieving the same capacity-approaching performance as LDPC block codes with iterative message-passing decoding. The first crucial property studied is the minimum free distance of LDPC convolutional code ensembles, an important parameter contributing to the error-correcting capability of the code. Here, asymptotic methods are used to form lower bounds on the ratio of the free distance to constraint length for several ensembles of asymptotically good, protograph-based LDPC convolutional codes. Further, it is shown that this ratio of free distance to constraint length for such LDPC convolutional codes exceeds the ratio of minimum distance to block length for corresponding LDPC block codes. Another interesting property of these codes is the way in which the structure affects the performance in the infamous error floor (which occurs at high signal to noise ratio) of the bit error rate curve. It has been suggested that “near-codewords” may be a significant factor affecting decoding failures of LDPC codes over an additive white Gaussian noise (AWGN) channel. A near-codeword is a sequence that satisfies almost all of the check equations. These nearcodewords can be associated with so-called ‘trapping sets’ that exist in the Tanner graph of a code. In the final major contribution of the thesis, trapping sets of protograph-based LDPC convolutional codes are analysed. Here, asymptotic methods are used to calculate a lower bound for the trapping set growth rates for several ensembles of asymptotically good protograph-based LDPC convolutional codes. This value can be used to predict where the error floor will occur for these codes under iterative message-passing decoding

Spatially Coupled Turbo-Like Codes

The focus of this thesis is on proposing and analyzing a powerful class of codes on graphs---with trellis constraints---that can simultaneously approach capacity and achieve very low error floor. In particular, we propose the concept of spatial coupling for turbo-like code (SC-TC) ensembles and investigate the impact of coupling on the performance of these codes. The main elements of this study can be summarized by the following four major topics. First, we considered the spatial coupling of parallel concatenated codes (PCCs), serially concatenated codes (SCCs), and hybrid concatenated codes (HCCs).We also proposed two extensions of braided convolutional codes (BCCs) to higher coupling memories. Second, we investigated the impact of coupling on the asymptotic behavior of the proposed ensembles in term of the decoding thresholds. For that, we derived the exact density evolution (DE) equations of the proposed SC-TC ensembles over the binary erasure channel. Using the DE equations, we found the thresholds of the coupled and uncoupled ensembles under belief propagation (BP) decoding for a wide range of rates. We also computed the maximum a-posteriori (MAP) thresholds of the underlying uncoupled ensembles. Our numerical results confirm that TCs have excellent MAP thresholds, and for a large enough coupling memory, the BP threshold of an SC-TC ensemble improves to the MAP threshold of the underlying TC ensemble. This phenomenon is called threshold saturation and we proved its occurrence for SC-TCs by use of a proof technique based on the potential function of the ensembles.Third, we investigated and discussed the performance of SC-TCs in the finite length regime. We proved that under certain conditions the minimum distance of an SC-TCs is either larger or equal to that of its underlying uncoupled ensemble. Based on this fact, we performed a weight enumerator (WE) analysis for the underlying uncoupled ensembles to investigate the error floor performance of the SC-TC ensembles. We computed bounds on the error rate performance and minimum distance of the TC ensembles. These bounds indicate very low error floor for SCC, HCC, and BCC ensembles, and show that for HCC, and BCC ensembles, the minimum distance grows linearly with the input block length.The results from the DE and WE analysis demonstrate that the performance of TCs benefits from spatial coupling in both waterfall and error floor regions. While uncoupled TC ensembles with close-to-capacity performance exhibit a high error floor, our results show that SC-TCs can simultaneously approach capacity and achieve very low error floor.Fourth, we proposed a unified ensemble of TCs that includes all the considered TC classes. We showed that for each of the original classes of TCs, it is possible to find an equivalent ensemble by proper selection of the design parameters in the unified ensemble. This unified ensemble not only helps us to understand the connections and trade-offs between the TC ensembles but also can be considered as a bridge between TCs and generalized low-density parity check codes

The Telecommunications and Data Acquisition Report

Archival reports are given on developments in programs managed by JPL's Office of Telecommunications and Data Acquisition (TDA), including space communications, radio navigation, radio science, ground-based radio and radar astronomy, and the Deep Space Network (DSN) and its associated Ground Communications Facility (GCF) in planning, supporting research and technology, implementation, and operations. Also included is TDA-funded activity at JPL on data and information systems and reimbursable DSN work performed for other space agencies through NASA. In the search for extraterrestrial intelligence (SETI), implementation and operations for searching the microwave spectrum are reported. Use of the Goldstone Solar System Radar for scientific exploration of the planets, their rings and satellites, asteroids, and comets are discussed