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Absorbing Sets and Error Floor Performance of the 5G New Radio Code

Masoome Otarinia

Published Nov 16, 2022• 0 citations
Publisher

Abstract

Low density parity check (LDPC) codes are a class of error correction codes. A structured class of LDPC codes known as quasi cyclic LDPC (QC-LDPC) codes have been selected for the data channels of the new generation of cellular communication standards known as 5G New Radio (NR). LDPC codes are generally decoded using iterative message passing algorithms that yield performance that is close to capacity. Previous research has shown that iterative decoders are vulnerable to certain combinatorial objects in the code’s structure resulting in an error floor, a typically abrupt change of slope in the frame error rate performance curve. These substructures are called absorbing sets; specifically, elementary absorbing sets are found to be particularly impactful for structured LDPC codes. The main focus of this thesis is investigating the decoder performance curve of the 5G NR code and identifying the error prone elementary absorbing sets that contribute to its error floor. In this thesis, we introduce the 5G NR QC-LDPC code and its characteristics; we talk about the measures taken in the transmitter to improve the code’s performance such as puncturing and shortening. We will see how practical limitations such as finite precision (quantization) of LLR values affects the performance in the error floor. We also devise an algorithm to identify all the (elementary) absorbing sets in the code.

DOI
10.7274/3j333200t9d

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Low density parity check (LDPC) codes are a class of error correction codes. A structured class of LDPC codes known as quasi cyclic LDPC (QC-LDPC) codes have been selected for the data channels of the new generation of cellular communication standards known as 5G New Radio (NR). LDPC codes are generally decoded using iterative message passing algorithms that yield performance that is close to capacity. Previous research has shown that iterative decoders are vulnerable to certain combinatorial objects in the code’s structure resulting in an error floor, a typically abrupt change of slope in the frame error rate performance curve. These substructures are called absorbing sets; specifically, elementary absorbing sets are found to be particularly impactful for structured LDPC codes. The main focus of this thesis is investigating the decoder performance curve of the 5G NR code and identifying the error prone elementary absorbing sets that contribute to its error floor. In this thesis, we introduce the 5G NR QC-LDPC code and its characteristics; we talk about the measures taken in the transmitter to improve the code’s performance such as puncturing and shortening. We will see how practical limitations such as finite precision (quantization) of LLR values affects the performance in the error floor. We also devise an algorithm to identify all the (elementary) absorbing sets in the code.
The data channels in the 5G New Radio use quasi-cyclic low-density parity check (LDPC) codes for error control. Such codes, when used with a practical quantizer and an iterative decoder, typically have an error floor that is strongly impacted by absorbing sets present in the code’s structure. This paper leverages the particular structure found in the 5G code to describe a simple means of identifying its (elementary) absorbing sets and describes an algorithm for finding them. It then uses simulation results to show which of the absorbing sets are responsible for the error floor performance for a particular choice of code and quantizer parameters.
Since their rediscovery in the late 1990s, low-density parity-check (LDPC) codes have rapidly become one the most important classes of error correcting codes. LDPC codes have a graphical description and are generally decoded iteratively. The graph of an iteratively decoded code's error rate performance versus the channel quality is typically divided into two regions. The error floor region appears at higher channel quality and is characterized by a more gradual decrease in error rate as channel quality improves. For iterative decoders the error floor is apparently determined by small structures within the code that are specific to the selected graphical description of the code and subtle details of the decoding algorithm. This thesis addresses several issues of LDPC error floors in the additive white Gaussian noise channel. First, we find the source of several well publicized error floors to be numerical issues within the message-passing decoder. We propose numerical improvements to several decoding formulations which substantially lower error floors. Next, we examine and refine error-floor prediction models. A time-variant linear state-space model is presented and analyzed. Graph theory is applied to analyze the small error-prone structures within the code. While we find our techniques encounter considerable problems in determining the error floors for iterative decoders with unrestricted dynamic range, their predictions are quite accurate when significant range restrictions are present. The final issue of LDPC error floors which we address relates to the minimum distance of quasi-cyclic (QC) LDPC codes. We derive improved minimum distance upper bounds for codes which are punctured and codes which contain a significant number of zeros in their protomatrices. These bound are then applied to a family of protograph-based LDPC codes standardized for use in deep-space communications. In the case of the rate-1/2 code, our upper bounds demonstrate that the previously promulgated ensemble lower bounds do not apply to QC constructions at block lengths beyond 4400 bits. The last part of this dissertation is a fairly separate piece of work on bounding the spectral radius of a large class of nonnegative matrices. We apply these bounds to digraphs and derive several associated equality conditions.</
Since their rediscovery in the late 1990s, low-density parity-check (LDPC) codes have rapidly become one the most important classes of error correcting codes. LDPC codes have a graphical description and are generally decoded iteratively. The graph of an iteratively decoded code's error rate performance versus the channel quality is typically divided into two regions. The error floor region appears at higher channel quality and is characterized by a more gradual decrease in error rate as channel quality improves. For iterative decoders the error floor is apparently determined by small structures within the code that are specific to the selected graphical description of the code and subtle details of the decoding algorithm. This thesis addresses several issues of LDPC error floors in the additive white Gaussian noise channel. First, we find the source of several well publicized error floors to be numerical issues within the message-passing decoder. We propose numerical improvements to several decoding formulations which substantially lower error floors. Next, we examine and refine error-floor prediction models. A time-variant linear state-space model is presented and analyzed. Graph theory is applied to analyze the small error-prone structures within the code. While we find our techniques encounter considerable problems in determining the error floors for iterative decoders with unrestricted dynamic range, their predictions are quite accurate when significant range restrictions are present. The final issue of LDPC error floors which we address relates to the minimum distance of quasi-cyclic (QC) LDPC codes. We derive improved minimum distance upper bounds for codes which are punctured and codes which contain a significant number of zeros in their protomatrices. These bound are then applied to a family of protograph-based LDPC codes standardized for use in deep-space communications. In the case of the rate-1/2 code, our upper bounds demonstrate that the previously promulgated ensemble lower bounds do not apply to QC constructions at block lengths beyond 4400 bits. The last part of this dissertation is a fairly separate piece of work on bounding the spectral radius of a large class of nonnegative matrices. We apply these bounds to digraphs and derive several associated equality conditions.</
Abstract In order to cater to the needs of the 5G standard, one of the prominent solutions in terms of high coding gain, high throughput and low power dissipation is the 5G NR LDPC (New Radio Low Density Parity Check) codes which have been adopted by 5G Standard. LDPC codes are the successful Third Generation Forward Error Correcting codes. An Efficient decoder is the key to the success of any Error Correcting Code. LDPC codes are one whose parity check matrix have a very few non zero entries. 5G NR LDPC codes are constructed by Protographs or base graphs and are block structured. This paper improves the performance of Layered Iterative minsum Approximation Message Passing decoding Algorithm using Rate Matching, Fixed Point Quantization and Offset minsum techniques. Rate Matching is to maintain the exact number of encoded blocks with in the TTI. Simulation of this algorithm with BPSK encoding over AWGN channel shows that BER of 10-3 gives Eb/No of~2 dB for the block length of 105.
It is well recognized that low-density parity-check (LDPC) codes can suffer from an error floor when decoded iteratively. This performance degradation is often attributed to the class of objects known as trapping sets. As a subset of the trapping set collection, there exists a class of graphical structures called the absorbing sets. An absorbing set is a combinatorially-defined object; in particular a fully absorbing set is stable under bit-flipping decoding. By construction, there can exist trapping sets that are not stable under such a decoder. As a result, for finite-precision, iterative decoding algorithms used over additive channels, absorbing sets can describe decoding errors more accurately than the broader class of trapping sets. In the first part of this thesis, we compute the normalized logarithmic asymptotic distributions of absorbing sets and fully absorbing sets, including elementary (fully) absorbing sets. We compare distributions of absorbing and trapping sets for representative code parameters of interest, and quantify the (lack of) discrepancies between the two. Good absorbing set properties are implied for known structured LDPC codes, including repeat accumulate codes and protograph-based constructions. Establishing the distribution of fully absorbing sets (especially when the discrepancy with the trapping set distribution is significant) allows one to further refine the estimates of the error rates under bit-flipping and related decoders.To reduce implementation complexity, the messages in a practical message passing decoder are necessarily quantized. Absorbing regions act as "decoding regions" around absorbing sets. In the second part of this thesis, we take a closer look at the interplay between quantization and absorbing regions. We provide a study of a range of quantization choices, the impact of quantization on the candidate absorbing regions, and derive guidelines for practical decoders. We show that, due to the non-linear dynamics of message passing decoders, coarser quantization may in fact perform better than finer quantization. Results of this type of work can be particularly useful in designing high performance decoders for very high-reliability storage systems, such as emerging data storage hard disk and solid state drives.
In contrast to the capacity approaching performance of iteratively decoded low-density parity check (LDPC) codes, many practical finite-length LDPC codes exhibit performance degradation, manifested in a so-called error floor. Previous work has linked this phenomenon to the presence of certain combinatorial structures within the Tanner graph representation of the code, termed absorbing sets. Absorbing sets are stable under the bit-flipping operations and have been shown to act as fixed points (¿absorbers¿) for a wider class of iterative decoding algorithms. Codes often possess absorbing sets whose size is smaller than the minimum distance: the smallest absorbing sets are deemed most detrimental culprits behind the error floor. This paper focuses on the elementary combinatorial bounds of the smallest (candidate) absorbing sets. For certain classes of practical codes we demonstrate the tightness of these bounds and show how can the structure of the code and the structure of the absorbing sets be utilized to increase the size of the smallest absorbing sets without compromising other code properties such as the node degrees and the girth. As such, this work provides a step towards a better code design by taking into account the combinatorial nature of fixed points of iterative decoding algorithms.