Finite-Length Scaling Laws for Spatially-Coupled LDPC Codes

This thesis concerns predicting the finite-length error-correcting performance of spatially-coupled low-density parity-check (SC-LDPC) code ensembles over the binary erasure channel. SC-LDPC codes are a very powerful class of codes; their use in practical communication systems, however, requires the system designer to specify a considerable number of code and decoder parameters, all of which affect both the code’s error-correcting capability and the system’s memory, energy, and latency requirements. Navigating the space of the associated trade-offs is challenging. The aim of the finite-length scaling laws proposed in this thesis is to facilitate code and decoder parameter optimization by providing a way to predict the code’s error-rate performance without resorting to Monte-Carlo simulations for each combination of code/decoder and channel parameters.First, we tackle the problem of predicting the frame, bit, and block error rate of SC-LDPC code ensembles over the binary erasure channel under both belief propagation (BP) decoding and sliding window decoding when the maximum number of decoding iterations is unlimited. The scaling laws we develop provide very accurate predictions of the error rates.Second, we derive a scaling law to accurately predict the bit and block error rate of SC-LDPC code ensembles with doping, a technique relevant for streaming applications for limiting the inherent rate loss of SC-LDPC codes. We then use the derived scaling law for code parameter optimization and show that doping can offer a way to achieve better transmission rates for the same target bit error rate than is possible without doping.Last, we address the most challenging (and most practically relevant) case where the maximum number of decoding iterations is limited, both for BP and sliding window decoding. The resulting predictions are again very accurate.Together, these contributions make finite-length SC-LDPC code and decoder parameter optimization via finite-length scaling laws feasible for the design of practical communication systems

A Refined Scaling Law for Spatially Coupled LDPC Codes Over the Binary Erasure Channel

We propose a refined scaling law to predict the finite-length performance in the waterfall region of spatially coupled low-density parity-check codes over the binary erasure channel. In particular, we introduce some improvements to the scaling law proposed by Olmos and Urbanke that result in a better agreement between the predicted and simulated frame error rate. We also show how the scaling law can be extended to predict the bit error rate performance.Comment: Paper accepted to IEEE Information Theory Workshop (ITW) 201

Finite-Length Scaling of Spatially Coupled LDPC Codes Under Window Decoding Over the BEC

We analyze the finite-length performance of spatially coupled low-density parity-check (SC-LDPC) codes under window decoding over the binary erasure channel. In particular, we propose a refinement of the scaling law by Olmos and Urbanke for the frame error rate (FER) of terminated SC-LDPC ensembles under full belief propagation (BP) decoding. The refined scaling law models the decoding process as two independent Ornstein-Uhlenbeck processes, in correspondence to the two decoding waves that propagate toward the center of the coupled chain for terminated SC-LDPC codes. We then extend the proposed scaling law to predict the performance of (terminated) SC-LDPC code ensembles under the more practical sliding window decoding. Finally, we extend this framework to predict the bit error rate (BER) and block error rate (BLER) of SC-LDPC code ensembles. The proposed scaling law yields very accurate predictions of the FER, BLER, and BER for both full BP and window decoding.Comment: Published in IEEE Transactions on Communications (Early Access). This paper was presented in part at the IEEE Information Theory Workshop (ITW), Visby, Sweden, August 2019 (arXiv:1904.10410

On Doped SC-LDPC Codes for Streaming

In streaming applications, doping improves the performance of spatially-coupled low-density parity-check (SC-LDPC) codes by creating reduced-degree check nodes in the coupled chain. We formulate a scaling law to predict the bit and block error rate of periodically-doped semi-infinite SC-LDPC code ensembles streamed over the binary erasure channel under sliding window decoding for a given finite component block length. The scaling law assumes that with some probability doping is equivalent to full termination and triggers two decoding waves; otherwise, decoding performs as if the coupled chain had not been doped at all. We approximate that probability and use the derived scaling laws to predict the error rates of SC-LDPC code ensembles in the presence of doping. The proposed scaling law provides accurate error rate predictions. We further use it to show that in streaming applications periodic doping can yield higher rates than periodic full termination for the same error-correcting performance

Finite-Length Scaling of SC-LDPC Codes With a Limited Number of Decoding Iterations

We propose four finite-length scaling laws to predict the frame error rate (FER) performance of spatially-coupled low-density parity-check codes under full belief propagation (BP) decoding with a limit on the number of decoding iterations and a scaling law for sliding window decoding, also with limited iterations. The laws for full BP decoding provide a choice between accuracy and computational complexity; a good balance between them is achieved by the law that models the number of decoded bits after a certain number of BP iterations by a time-integrated Ornstein-Uhlenbeck process. This framework is developed further to model sliding window decoding as a race between the integrated Ornstein-Uhlenbeck process and an absorbing barrier that corresponds to the left boundary of the sliding window. The proposed scaling laws yield accurate FER predictions