Sparse Codes on Graphs with Convolutional Code Constraints

Abstract: Modern coding theory is based on the foundation of the sparse codes on graphs, such as the low-density parity-check (LDPC) codes, and the turbo-like codes (TCs) with component convolutional codes. The success of the LDPC codes and the TCs lies in their ability to perform low-complexity iterative message passing decoding procedures. The iterative message passing decoders that exchange messages probabilities, or beliefs, within the code graph are known as the belief propagation (BP) decoders. The BP decoders are sub-optimal, whereas maximum-a-posteriori (MAP) decoders for these codes are computationally infeasible. These codes can be optimized for their BP decoding performance, which improves their error rate performance in the waterfall region at the cost of a performance loss in the error-floor region. On the contrary, optimizing these codes for the MAP performance results in an improved performance in the error-floor region at the expense of a degraded performance in the waterfall region.In practice, the BP decoding performance of the LDPC codes and the TCs, in the asymptotic block length regime, is determined by computing their BP decoding thresholds from the density evolution (DE), or the extrinsic information transfer (EXIT) chart techniques. The MAP decoding thresholds can be obtained with an application of the area theorem to the BP decoder performance. The graphs of the LDPC codes and the TCs are optimized for the BP, and the MAP decoding performance by using their decoding thresholds. For very large block lengths, spatially coupled (SC) versions of LDPC codes, and the TCs-which are optimized for MAP decoding performance- were shown to achieve excellent BP decoding performance in both the waterfall and the error-floor region, thanks to the threshold saturation. However, the BP decoding performance of these spatially coupled codes suffer from a high error-floor at a moderate to short code block length.BP and MAP decoding thresholds of TCs on a binary erasure channel (BEC) have previously been investigated via the DE analysis. The capacity achieving SC-TCs were determined, where the underlying TCs were optimized for the BP and MAP performance. The TCs ensembles, parallel concatenated codes, serially concatenated codes, braided convolutional codes, and hybrid concatenated codes, with varying component convolutional codes strengths were considered in these investigations.This thesis focuses on investigating the BP decoding performance of SC-TCs- which were earlier investigated for the BEC- on an additive white Gaussian noise (AWGN) channel. Furthermore, the problem of high error-floor of SC-TCs for short to moderate block lengths is investigated and addressed with a design of an optimized convlutional permutor in constructing the SC-TCs. Finally, the connection of the TCs and the LDPC codes is explored by introducing a family of convolutional codes (CC) based generalized LDPC codes (GLDPCs). These research areas are summarized under the following three topics.In the first topic, we have computed the iterative decoding thresholds of SC-TCs on the AWGN channel via the Monte Carlo density evolution (MC-DE) methods. The MC-DE methods are time consuming, which has motivated us to introduce an efficient alternative that predicts the AWGN thresholds of SC-TCs with the knowledge of their BEC thresholds. The results show that the estimated thresholds via the MC-DE method and the predicted thresholds are very close for the capacity achieving randomly punctured SC-TCs. For the high rate uncoupled TCs, which are obtained by randomly puncturing their mother code, the predicted thresholds are improved by incorporating the estimated AWGN threshold of the mother code ensemble into the threshold prediction method.In the second topic, we have introduced the design of a single block-wise periodic time-varying convolutional permutor to construct the SC-TCs. The convolutional permutor is designed by applying the unwrapping procedure to an optimized block permutor, which optimize the bit error rate (BER) performance of a TC in an error-floor region. We showed that a convolutional permutor obtained via the unwrapping procedure inherits the properties of its parent permutor. Due to this reason, the BER performance of block-wise periodically time-varying convolutional permutor based SC-SCCs does not suffer from a high error-floor problem at short block lengths, which was demonstrated through the simulation results.In the third topic, we have introduced the families of regular and irregular CC-GLDPCs. The CC-GLDPCs enabled us to connect TCs and LDPC codes in terms of their graph structures. The BEC thresholds and the minimum distance properties of the regular CC-GLDPCs were compared to the regular LDPC codes. Furthermore, we performed an exhaustive grid search using the BEC thresholds of the class of CC-GLDCPs, and determined the design configurations of optimized CC-GLDPCs. The results suggest that, for regular graphs, it is possible to find a sparser CC-GLDPCs than the LDPC codes at the expense of a slightly negligible loss in the performance. Furthermore, the BP optimized CC-GLDPC is observed to have a better BP and MAP thresholds than the turbo codes on the BEC.