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arXiv:2402.06814 · International Conference on Communications

High-Rate Fair-Density Parity-Check Codes

Hessam Mahdavifar

Published Jun 9, 2024• 4 citations
arXivHTMLPDF

Abstract

We introduce fair-density parity-check (FDPC) codes targeting high-rate applications. In particular, we start with a base parity-check matrix$H_{b}$of dimension$2\sqrt{n}\times n$, where$n$is the code block length, and the number of ones in each row and column of$H_{b}$is equal to$\sqrt{n}$and 2, respectively. We propose a deterministic combinatorial method for picking the base matrix$H_{b}$, assuming$n=4t^{2}$for some integer$t\geqslant 2$. We then extend this by obtaining permuted versions of$H_{b}$(e.g., via random permutations of its columns) and stacking them on top of each other leading to codes of dimension$k\geqslant n-2s\sqrt{n}+s$, for some$s\geqslant 2$, referred to as order-s FDPC codes. We propose methods to explicitly characterize and bound the weight distribution of the new codes and utilize them to derive union-type approximate upper bounds on their error probability under Maximum Likelihood (ML) decoding. For the binary erasure channel (BEC), we demonstrate that the approximate ML bound of FDPC codes closely follows the random coding upper bound (RCU) for a wide range of channel parameters. Also, remarkably, FDPC codes, under the low-complexity min-sum decoder, improve upon 5G-LDPC codes for transmission over the binary-input additive white Gaussian noise (B-AWGN) channel by almost 0.5dB (for$n=1024$, and rate = 0.878). Furthermore, we propose a new decoder as a combination of weighted min-sum message-passing (MP) decoding algorithm together with a new progressive list (PL) decoding component, referred to as the MP-PL decoder, to further boost the performance of FDPC codes. This paper opens new avenues for a fresh investigation of new code constructions and decoding algorithms in high-rate regimes suitable for ultra-high throughput (high-frequency/optical) applications.

DOI
10.1109/icc51166.2024.10622808

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Low-density parity-check (LDPC) codes have been one of the most popular error correction candidates in modern communication and data storage systems for the past three decades due to their capacity-approaching error correction capability under the Belief Propagation (BP) decoding algorithms. In this dissertation, we investigate three topics related to efficient and effective decoding algorithms and the code design of LDPC codes. In the first part of this dissertation, we propose a methodology with the help of artificial intelligence to design Finite Alphabet Iterative Decoders (FAID) for LDPC codes, which are not only of great theoretical but also practical importance. This model-driven neural network (NN) framework unfolds the decoding iterations, introduces quantized neural networks (QNN) to model the FAIDs, and proposes a new cost function, namely, bit error rate (BER), to address the inefficiency in discrete-output channel scenarios. The low precision activations in the QNN and quantization in the cost functions cause a critical issue that their gradients vanish almost everywhere, making it difficult to use classical backward propagation. We design and leverage straight-through estimators as surrogate gradients to tackle this issue and provide a joint training scheme. These QNN-aided FAIDs, are capable of surpassing the floating-point BP algorithms, with much lower complexity and faster convergence. The QNN-aided approach is general, as it can be applied to both continuous and discrete-output channels. In the second part of this dissertation, we propose a new code construction of LDPC codes based on the conventional parity-check matrices of Reed-Solomon (RS) codes. We investigate the girth and cycle structure and introduce an efficient searching algorithm to reduce short cycles in their Tanner graphs and to construct LDPC codes with a girth of at least eight. We use masking to further reduce the number of short cycles and to adjust the column and row weights of the parity-check matrices of the constructed codes for performance improvement. Our designs are systematic and flexible, which balance the requirements of good waterfall performance and low error-floor while maintaining a structure that facilitates implementation. In the third part of this dissertation, we investigate binary quasi-cyclic (QC) LDPC codes and their capability to correct phased bursts of erasures. We prove that the peeling algorithm when applied to such parity-check matrices of column weight two can correct all erasures that can be corrected by a maximum likelihood (ML) decoder. We show that by modifying the parity-check matrices of column weight two and globally coupling them, the erasure correcting capability can be further enhanced. For the Additive White Gaussian Noise (AWGN) channel, we design QC LDPC codes with parity-check matrices of column weight three or more that can correct phased bursts of erasures and perform well over the AWGN channel. Our analysis of QC-LDPC codes with parity-check matrices of column weight two is rather comprehensive as we determine, for all such codes, their dimensions, minimum Hamming distances, and their capabilities to correct phased bursts.
A unified approach to search for and optimize codes determined by their sparse parity-check matrices is presented. Replacing the nonzero elements of a binary parity-check matrix (the base or parent matrix) either by circulants or by companion matrices of elements from a finite field GF(2m), we obtain quasi-cyclic low-density parity-check (LDPC) block codes and binary images of nonbinary LDPC block codes, respectively. By substituting monomials of a formal variable D, we obtain the polynomial description of an LDPC convolutional code. A set of performance measures applicable to different classes of LDPC codes is considered, and a greedy algorithm for code performance optimization is presented. The heart of the new optimization algorithm is a fast procedure for searching for LDPC codes with large girth of their Tanner graphs. For a few classes of LDPC codes, examples of codes combining good error-correcting performance with compact representation are obtained. In particular, we present optimized convolutional LDPC codes and conclude that the LDPC block codes are still superior to their convolutional counterparts if both decoding complexity and coding delay are considered. Moreover, a specific channel model can easily be embedded into the optimization loop. Thereby, the code can be optimized for a specific channel. The efficiency of such an optimization is demonstrated via an example of faster than Nyquist (FTN) signaling using LDPC codes. The FTN strategy combined with a rate R = 1/2 LDPC code of length 64800 optimized for effective data rate R = 3/4 gains more than 0.5 dB compared with the standard LDPC codes of the same rate and length. The obtained gain corresponds to transmission at the capacity of the binary input additive white Gaussian noise channel. In most numerical examples, we consider codes with bidiagonal structure of the parity-check matrix. This restriction preserves low encoding complexity and allows fair comparison with codes selected for communication standards.
High-rate irregular Generalized LDPC (GLDPC) codes with low-complexity constituent codes are studied. The main area of focus is the optimization of the underlying irregular binary LDPC codes and their matching with constituent codes. Finite length random coding bound on the maximum-likelihood decoding error probability of a random ensemble of irregular GLDPC codes is derived and compared with a similar bound for non-binary (NB) LDPC codes over small alphabets.A new algorithm for assigning columns of the parity-check matrix of a constituent code to nonzero elements of a base matrix of the quasi-cyclic (QC)-GLDPC code is presented. Examples of the constructed QC-GLDPC codes are given. The simulation results for frame error rate (FER) performance of the belief-propagation (BP) decoding for the QC-GLDPC codes are presented and compared with the same performance of NB QC-LDPC codes over small alphabets and binary LDPC codes in the WiFi and 5G standards.Both GLDPC codes and NB LDPC codes with sufficiently low decoding complexity are shown to outperform their binary LDPC counterparts, while GLDPC codes appear to be more advantageous in terms of decoding complexity.