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Short LDPC Codes with Low Undetected Frame Error Rate: Post-Decoding Error Detection Criteria

Yilmaz, Fatmanur

Published Nov 1, 2024• 0 citations
OpenAlex

Abstract

Low-density parity-check (LDPC) codes have become a standard in modern communication systems thanks to their near-optimal error correction performance, which approaches Shannon's channel capacity as block length increases. However, in scenarios requiring short block lengths with low undetected frame error rate (UFER)-such as command and control data transmission in satellite communications-using only iterative decoders fails to achieve a balance between undetected and total frame error rates (FERs). This thesis addresses these challenges by investigating post-decoding error detection methods that enhance iterative LDPC decoding to achieve target FERs for both total and undetected errors. Specifically, we work with (128, 64) LDPC codes and explore two novel post-decoding techniques: bounded angle iterative decoding and single bit flip (SBF) list decoding, a variation of SBF decoding. These methods are applied to the iterative LDPC decoder output to manage the trade-off between total and undetected FERs effectively. Our results demonstrate that bounded angle iterative decoding effectively reduces UFER in codes with a large minimum distance with maintaining an acceptable increase in total frame error rate (TFER). Meanwhile, SBF list decoding mitigates the UFER increase introduced by standard SBF decoding, while retaining the significant TFER improvement that SBF decoding offers.

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Low-density parity-check (LDPC) codes have become a standard in modern communication systems thanks to their near-optimal error correction performance, which approaches Shannon's channel capacity as block length increases. However, in scenarios requiring short block lengths with low undetected frame error rate (UFER)-such as command and control data transmission in satellite communications-using only iterative decoders fails to achieve a balance between undetected and total frame error rates (FERs). This thesis addresses these challenges by investigating post-decoding error detection methods that enhance iterative LDPC decoding to achieve target FERs for both total and undetected errors. Specifically, we work with (128, 64) LDPC codes and explore two novel post-decoding techniques: bounded angle iterative decoding and single bit flip (SBF) list decoding, a variation of SBF decoding. These methods are applied to the iterative LDPC decoder output to manage the trade-off between total and undetected FERs effectively. Our results demonstrate that bounded angle iterative decoding effectively reduces UFER in codes with a large minimum distance with maintaining an acceptable increase in total frame error rate (TFER). Meanwhile, SBF list decoding mitigates the UFER increase introduced by standard SBF decoding, while retaining the significant TFER improvement that SBF decoding offers.
Differently from bounded-distance decoders used for algebraic codes, iterative decoders used for low-density parity-check (LDPC) codes are not characterized by a deterministic decoding radius. Therefore, the error rates of LDPC-coded transmissions are usually estimated heuristically through simulations. This is adequate for many applications like wireless communications, where a frame error rate (FER) in the order of 10-6or higher is usually targeted. However, lower values of FER can barely be assessed through simulations, and this limits the use of LDPC codes in applications requiring a lower FER, like optical communications and code-based cryptography. In this paper we introduce and study a version of the classic bit flipping (BF) decoder for which we are able to devise and develop a theoretical characterization of the FER. In addition, we consider a two-iteration hard-decision decoder for LDPC codes derived from BF, and discuss its error rate performance. Our results are validated through numerical simulations.
A method for estimating the performance of low-density parity-check (LDPC) codes decoded by hard-decision iterative decoding algorithms on binary symmetric channels (BSCs) is proposed. Based on the enumeration of the smallest weight error patterns that cannot be all corrected by the decoder, this method estimates both the frame error rate (FER) and the bit error rate (BER) of a given LDPC code with very good precision for all crossover probabilities of practical interest. Through a number of examples, we show that the proposed method can be effectively applied to both regular and irregular LDPC codes and to a variety of hard-decision iterative decoding algorithms. Compared with the conventional Monte Carlo simulation, the proposed method has a much smaller computational complexity, particularly for lower error rates.
Error control coding is an essential part of modern communications systems. LDPC codes have been demonstrated to offer performance near the fundamental limits of channels corrupted by random noise. Optimal maximum likelihood decoding of LDPC codes is too complex to be practically useful even at short block lengths and so a graph-based message passing decoder known as the belief propagation algorithm is used instead. In fact, on graphs without closed paths known as cycles the iterative message passing decoding is known to be optimal and may converge in a single iteration, although identifying the message update schedule which allows single-iteration convergence is not trivial. At finite block lengths graphs without cycles have poor minimum distance properties and perform poorly even under optimal decoding. LDPC codes with large block length have been demonstrated to offer performance close to that predicted for codes of infinite length, as the cycles present in the graph are quite long. In this thesis, LDPC codes of shorter length are considered as they offer advantages in terms of latency and complexity, at the cost of performance degradation from the increased number of short cycles in the graph. For these shorter LDPC codes, the problems considered are: \n \nFirst, improved construction of structured and unstructured LDPC code graphs of short length with a view to reducing the harmful effects of the cycles on error rate performance, based on knowledge of the decoding process. Structured code graphs are particularly interesting as they allow benefits in encoding and decoding complexity and speed. Secondly, the design and construction of LDPC codes for the block fading channel, a particularly challenging scenario from the point of view of error control code design. Both established and novel classes of codes for the channel are considered. Finally the decoding of LDPC codes by the belief propagation algorithm is considered, in particular the scheduling of messages passed in the iterative decoder. A knowledge-aided approach is developed based on message reliabilities and residuals to allow fast convergence and significant improvements in error rate performance.
A method for estimating the performance of low-density parity-check (LDPC) codes decoded by hard-decision iterative decoding algorithms on binary symmetric channels (BSC) is proposed. Based on the enumeration of the smallest weight error patterns that cannot be all corrected by the decoder, this method estimates both the frame error rate (FER) and the bit error rate (BER) of a given LDPC code with very good precision for all crossover probabilities of practical interest. Through a number of examples, we show that the proposed method can be effectively applied to both regular and irregular LDPC codes and to a variety of hard-decision iterative decoding algorithms. Compared with the conventional Monte Carlo simulation, the proposed method has a much smaller computational complexity, particularly for lower error rates.
The performance of low-density parity-check (LDPC) codes decoded by hard-decision iterative decoding algorithms can be accurately estimated if the weight J and the number |Ej| of the smallest error patterns that cannot be corrected by the decoder are known. To obtain J and |Ej|, one would need to perform the direct enumeration of error patterns with weight i les J. The complexity of enumeration increases exponentially with J, essentially as nJ, where n is the code block length. In this paper, we approximate J and |Ej| by enumerating and testing the error patterns that are subsets of short cycles in the code's Tanner graph. This reduces the computational complexity by several orders of magnitude compared to direct enumeration, making it possible to estimate the error rates for almost any practical LDPC code. To obtain the error rate estimates, we propose an algorithm that progressively improves the estimates as larger cycles are enumerated. Through a number of examples, we demonstrate that the proposed method can accurately estimate both the bit error rate (BER) and the frame error rate (FER) of regular and irregular LDPC codes decoded by a variety of hard-decision iterative decoding algorithms.
A modification to the usual iterative decoding algorithm for LDPC codes, called bounded angle iterative (BA-I) decoding, is introduced. The modified decoder erases codewords detected during iterations that fall outside a maximum decoding angle with respect to the received observation. The new algorithm is applicable in scenarios that demand a very low undetected error rate but require short LDPC codes that are too vulnerable to undetected errors when the usual iterative decoding algorithm is used. BA-I decoding provides a means of reducing the maximum undetected error rate for short LDPC codes significantly, by incorporating a simple extra condition into the iterative decoder structure without redesigning the code. The reduction in undetected error rate comes at a price of increasing the threshold signal-to-noise ratio (SNR) required for achieving a good overall error rate, but this increase in channel threshold can be minimized by allowing the decoderpsilas maximum decoding angle to vary with SNR.