Sudarsan V. S. Ranganathan

Short-blocklength non-binary LDPC codes with feedback-dependent incremental transmissions

One advantage of feedback in a point-to-point memoryless channel is the reduction of the average blocklength required to approach capacity. This paper presents a communication system with feedback that uses carefully designed non-binary LDPC (NB-LDPC) codes and incremental transmissions to achieve 92-94% of the idealized throughput of rate-compatible sphere-packing with maximum-likelihood decoding (RCSP-ML) for average blocklengths of 150-450 bits. The system uses active feedback by carefully selecting each bit of additional incremental information to improve the reliability of the least reliable variable node. The system uses post processing in the decoder to further improve performance. The average blocklengths of 150-450 bits are small enough that feedback provides a throughput advantage but also large enough that overhead that might be associated with transmitter confirmation is more easily tolerated.

On the girth of (3,L) quasi-cyclic LDPC codes based on complete protographs

We consider the problem of constructing (3,L) quasi-cyclic low-density parity-check (LDPC) codes from complete protographs. A complete protograph is a small bipartite graph with two disjoint vertex sets such that every vertex in the variable-node set is connected to every vertex in the check-node set by a unique edge. This paper analyzes the required lifting factor for achieving girths of six or eight in the resulting quasi-cyclic codes with constraints on lifting. The required lifting factors provide lower bounds on the block-length of such codes.

Design of high-rate irregular non-binary LDPC codes using algorithmic stopping-set cancellation

Following Poulliat et al.s design of (2, dc) nonbinary LDPC (NB-LDPC) codes, this paper designs high-rate irregular NB-LDPC codes by addressing the problem of minimum symbol distance. The design procedure first identifies all stopping sets up to weight five in an LDPC code and enumerates them via a message passing algorithm. For each identified stopping set, careful labeling forces its corresponding parity-check sub-matrix to be full rank, thereby preventing the stopping set from being a sub-code and ensuring a minimum distance of at least six symbols. Simulation results for codes designed through this procedure show a significant improvement in the error-floor region over randomized labeling.

Optimizing Transmission Lengths for Limited Feedback With Nonbinary LDPC Examples

This paper presents a general approach for optimizing the number of symbols in increments (packets of incremental redundancy) in a feedback communication system with a limited number of increments. This approach is based on a tight normal approximation on the rate for successful decoding. Applying this approach to a variety of feedback systems using nonbinary (NB) low-density parity-check (LDPC) codes shows that greater than 90% of capacity can be achieved with average blocklengths fewer than 500 transmitted bits. One result is that the performance with ten increments closely approaches the performance with an infinite number of increments. The paper focuses on binary-input additive-white Gaussian noise (BI-AWGN) channels but also demonstrates that the normal approximation works well on examples of fading channels as well as high-SNR AWGN channels that require larger QAM constellations. This paper explores both variable-length feedback codes with termination (VLFT) and the more practical variable length feedback (VLF) codes without termination that require no assumption of noiseless transmitter confirmation. For VLF, we consider both a two-phase scheme and CRC-based scheme.

Feedback systems using non-binary LDPC codes with a limited number of transmissions

One advantage of incremental transmissions with feedback in point-to-point memoryless channels is a reduction in average blocklength required to approach capacity. This paper optimizes the size of each incremental transmission for non-binary (NB) LDPC codes to maximize throughput in VLFT and two-phase VLF settings. The optimization problem uses an approximation based on the inverse-Gaussian p.d.f. of the blocklength required for successful decoding. By using the optimized incremental transmission lengths (with an average blocklength of less than 500 bits), NB-LDPC codes for VLFT setting limited to 5 transmissions achieve a throughput greater than 96% of that obtained by an unlimited-transmission VLFT scheme with the same average blocklength. With a similar average blocklength, a two-phase VLF system limited to five transmissions (with optimized lengths) using the binary image of NB-LDPC codes achieves greater than 90% of the capacity of binary-input AWGN channel with SNR=2 dB. Two-phase VLF does not match the throughput of VLFT, but it is more practical than VLFT because it does not assume noiseless transmitter confirmation.

Sequential differential optimization of incremental redundancy transmission lengths: An example with tail-biting convolutional codes

This paper applies the sequential differential optimization (SDO) algorithm to optimize the transmission lengths of incremental redundancy for a 1024-state tail-biting convolutional code. The tail-biting reliability-output Viterbi algorithm is used to determine whether to inform the transmitter that a message has been successfully received or to request that the transmitter provide additional convolutional code bits. In order to maximize the average throughput, SDO is used to determine the rate of the initial codeword and the number of bits of incremental redundancy to be sent in each increment. With the help of SDO, this paper demonstrates a system that achieves 86.3 percent of the binary-input AWGN capacity (for SNR 2 dB) with an average blocklength of 115.5 symbols.

Some results on spatially coupled protograph LDPC codes

Spatially coupled low-density parity-check block codes (SC-LDPC-BCs) are a class of LDPC codes in which certain ensembles achieve capacity under iterative belief propagation decoding. Prior work has considered these codes mostly from a protograph perspective. Following this viewpoint, our work presents new protographs for the design of protograph-based SC-LDPC-BCs. A reciprocal channel approximation (RCA) analysis of the terminated protographs yields the asymptotic performance limits of the ensembles in terms of their iterative decoding thresholds. Our protographs possess the properties of linear minimum distance growth rate and excellent thresholds. We consider the following channels in our work: binary-input additive white Gaussian noise channel (BI-AWGNC), binary erasure channel (BEC), and binary symmetric channel (BSC). Our work focuses on obtaining good designs, with two classes of ensembles being asymptotically regular, for all the three channels while minimizing rate loss.

Approaching capacity using incremental redundancy without feedback

Variable-length codes with incremental redundancy controlled by feedback allow a system to approach capacity with short average blocklengths and thus relatively low-complexity decoders. This paper shows how to use those same variable-length codes with incremental redundancy to approach capacity without feedback. The general principle is to provide a common pool of redundancy that can be accessed by exactly the variable-length codes that need it. We provide example implementations using both regular and irregular low-density generator matrix (LDGM) codes to provide this common pool of redundancy, utilizing the inter-frame coding approach that Zeineddine and Mansour used to combat rate variation due to fading in broadcast transmissions. Obtaining the LDGM degree distributions requires a new design methodology involving differential evolution for a generalized peeling decoder. Monte-Carlo simulations using a 2dB binary-input additive white Gaussian noise channel confirm the feasibility of this new approach. For a frame error rate of 10−3, the irregular LDGM code achieves 96% of the throughput of the corresponding feedback system.

An information density approach to analyzing and optimizing incremental redundancy with feedback

This paper uses a case study of a tail-biting convolutional code (with successful decoding indicated by the reliability output Viterbi algorithm) to present an information density approach for analyzing and optimizing the throughput of systems using incremental redundancy controlled by feedback. Polyan-skiys normal approximation combined with a linear model for the information gap of a rate-compatible code family provides a simple and accurate characterization of the behavior of feedback systems employing practical codes, such as convolutional or low-density parity-check codes. Especially for short message lengths on the order of k < 50 message bits, the newly proposed model is more accurate than Vakilinias model in which the rate of first successful decoding has a Gaussian probability density function.

Design of improved quasi-cyclic protograph-based Raptor-like LDPC codes for short block-lengths

Protograph-based Raptor-like low-density parity-check codes (PBRL codes) are a recently proposed family of easily encodable and decodable rate-compatible LDPC (RC-LDPC) codes. These codes have an excellent iterative decoding threshold and performance across all design rates. PBRL codes designed thus far, for both long and short block-lengths, have been based on optimizing the iterative decoding threshold of the protograph of the RC code family at various design rates. In this work, we propose a design method to obtain better quasi-cyclic (QC) RC-LDPC codes with PBRL structure for short block-lengths (of a few hundred bits). We achieve this by maximizing an upper bound on the minimum distance of any QC-LDPC code that can be obtained from the protograph of a PBRL ensemble. The obtained codes outperform the original PBRL codes at short block-lengths by significantly improving the error floor behavior at all design rates. Furthermore, we identify a reduction in complexity of the design procedure, facilitated by the general structure of a PBRL ensemble.