Spatially-Coupled Random Access on Graphs

In this paper we investigate the effect of spatial coupling applied to the recently-proposed coded slotted ALOHA (CSA) random access protocol. Thanks to the bridge between the graphical model describing the iterative interference cancelation process of CSA over the random access frame and the erasure recovery process of low-density parity-check (LDPC) codes over the binary erasure channel (BEC), we propose an access protocol which is inspired by the convolutional LDPC code construction. The proposed protocol exploits the terminations of its graphical model to achieve the spatial coupling effect, attaining performance close to the theoretical limits of CSA. As for the convolutional LDPC code case, large iterative decoding thresholds are obtained by simply increasing the density of the graph. We show that the threshold saturation effect takes place by defining a suitable counterpart of the maximum-a-posteriori decoding threshold of spatially-coupled LDPC code ensembles. In the asymptotic setting, the proposed scheme allows sustaining a traffic close to 1 [packets/slot].Comment: To be presented at IEEE ISIT 2012, Bosto

Generalized Stability Condition for Generalized and Doubly-Generalized LDPC Codes

In this paper, the stability condition for low-density parity-check (LDPC) codes on the binary erasure channel (BEC) is extended to generalized LDPC (GLDPC) codes and doublygeneralized LDPC (D-GLDPC) codes. It is proved that, in both cases, the stability condition only involves the component codes with minimum distance 2. The stability condition for GLDPC codes is always expressed as an upper bound to the decoding threshold. This is not possible for D-GLDPC codes, unless all the generalized variable nodes have minimum distance at least 3. Furthermore, a condition called derivative matching is defined in the paper. This condition is sufficient for a GLDPC or DGLDPC code to achieve the stability condition with equality. If this condition is satisfied, the threshold of D-GLDPC codes (whose generalized variable nodes have all minimum distance at least 3) and GLDPC codes can be expressed in closed form.Comment: 5 pages, 2 figures, to appear in Proc. of IEEE ISIT 200

Decomposition Methods for Large Scale LP Decoding

When binary linear error-correcting codes are used over symmetric channels, a relaxed version of the maximum likelihood decoding problem can be stated as a linear program (LP). This LP decoder can be used to decode error-correcting codes at bit-error-rates comparable to state-of-the-art belief propagation (BP) decoders, but with significantly stronger theoretical guarantees. However, LP decoding when implemented with standard LP solvers does not easily scale to the block lengths of modern error correcting codes. In this paper we draw on decomposition methods from optimization theory, specifically the Alternating Directions Method of Multipliers (ADMM), to develop efficient distributed algorithms for LP decoding. The key enabling technical result is a "two-slice" characterization of the geometry of the parity polytope, which is the convex hull of all codewords of a single parity check code. This new characterization simplifies the representation of points in the polytope. Using this simplification, we develop an efficient algorithm for Euclidean norm projection onto the parity polytope. This projection is required by ADMM and allows us to use LP decoding, with all its theoretical guarantees, to decode large-scale error correcting codes efficiently. We present numerical results for LDPC codes of lengths more than 1000. The waterfall region of LP decoding is seen to initiate at a slightly higher signal-to-noise ratio than for sum-product BP, however an error floor is not observed for LP decoding, which is not the case for BP. Our implementation of LP decoding using ADMM executes as fast as our baseline sum-product BP decoder, is fully parallelizable, and can be seen to implement a type of message-passing with a particularly simple schedule.Comment: 35 pages, 11 figures. An early version of this work appeared at the 49th Annual Allerton Conference, September 2011. This version to appear in IEEE Transactions on Information Theor

Stability of Iterative Decoding of Multi-Edge Type Doubly-Generalized LDPC Codes Over the BEC

Using the EXIT chart approach, a necessary and sufficient condition is developed for the local stability of iterative decoding of multi-edge type (MET) doubly-generalized low-density parity-check (D-GLDPC) code ensembles. In such code ensembles, the use of arbitrary linear block codes as component codes is combined with the further design of local Tanner graph connectivity through the use of multiple edge types. The stability condition for these code ensembles is shown to be succinctly described in terms of the value of the spectral radius of an appropriately defined polynomial matrix.Comment: 6 pages, 3 figures. Presented at Globecom 2011, Houston, T

On generalized LDPC codes for ultra reliable communication

Ultra reliable low latency communication (URLLC) is an important feature in future mobile communication systems, as they will require high data rates, large system capacity and massive device connectivity [11]. To meet such stringent requirements, many error-correction codes (ECC)s are being investigated; turbo codes, low density parity check (LDPC) codes, polar codes and convolutional codes [70, 92, 38], among many others. In this work, we present generalized low density parity check (GLDPC) codes as a promising candidate for URLLC. Our proposal is based on a novel class of GLDPC code ensembles, for which new analysis tools are proposed. We analyze the trade-o_ between coding rate and asymptotic performance of a class of GLDPC codes constructed by including a certain fraction of generalized constraint (GC) nodes in the graph. To incorporate both bounded distance (BD) and maximum likelihood (ML) decoding at GC nodes into our analysis without resorting to multi-edge type of degree distribution (DD)s, we propose the probabilistic peeling decoding (P-PD) algorithm, which models the decoding step at every GC node as an instance of a Bernoulli random variable with a successful decoding probability that depends on both the GC block code as well as its decoding algorithm. The P-PD asymptotic performance over the BEC can be efficiently predicted using standard techniques for LDPC codes such as Density evolution (DE) or the differential equation method. We demonstrate that the simulated P-PD performance accurately predicts the actual performance of the GLPDC code under ML decoding at GC nodes. We illustrate our analysis for GLDPC code ensembles with regular and irregular DDs. This design methodology is applied to construct practical codes for URLLC. To this end, we incorporate to our analysis the use of quasi-cyclic (QC) structures, to mitigate the code error floor and facilitate the code very large scale integration (VLSI) implementation. Furthermore, for the additive white Gaussian noise (AWGN) channel, we analyze the complexity and performance of the message passing decoder with various update rules (including standard full-precision sum product and min-sum algorithms) and quantization schemes. The block error rate (BLER) performance of the proposed GLDPC codes, combined with a complementary outer code, is shown to outperform a variety of state-of-the-art codes, for URLLC, including LDPC codes, polar codes, turbo codes and convolutional codes, at similar complexity rates.Programa Oficial de Doctorado en Multimedia y ComunicacionesPresidente: Juan José Murillo Fuentes.- Secretario: Matilde Pilar Sánchez Fernández.- Vocal: Javier Valls Coquilla

EBP-GEXIT charts over the binary-input AWGN channel for generalized and doubly-generalized LDPC codes

This work proposes a tractable evaluation of the maximum a posteriori (MAP) threshold of sparse-graph ensembles, by using an approximation for the extended belief propagation generalized extrinsic information transfer (EBP-GEXIT) function, first proposed by Measson et al. The approximation allows to find a MAP threshold in such numerically involved cases as the binary-input additive white Gaussian noise (AWGN) channel, graph ensembles with general component codes and/or irregularities. The paper contains examples of estimations of the MAP thresholds in the case of irregular low-density parity-check (LDPC), generalized LDPC, and doubly generalized LDPC codes ensembles. Our estimations are confirmed by numerical simulations

Design of LDPC Code Ensembles with Fast Convergence Properties

The design of low-density parity-check (LDPC) code ensembles optimized for a finite number of decoder iterations is investigated. Our approach employs EXIT chart analysis and differential evolution to design such ensembles for the binary erasure channel and additive white Gaussian noise channel. The error rates of codes optimized for various numbers of decoder iterations are compared and it is seen that in the cases considered, the best performance for a given number of decoder iterations is achieved by codes which are optimized for this particular number. The design of generalized LDPC (GLDPC) codes is also considered, showing that these structures can offer better performance than LDPC codes for low-iteration-number designs. Finally, it is illustrated that LDPC codes which are optimized for a small number of iterations exhibit significant deviations in terms of degree distribution and weight enumerators with respect to LDPC codes returned by more conventional design tools.Comment: 6 pages, 5 figures, Submitted to the 3rd International Black Sea Conference on Communications and Networking (IEEE BlackSeaCom 2015