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首页> 外文期刊>IEEE Transactions on Information Theory >A Combinatorial Methodology for Optimizing Non-Binary Graph-Based Codes: Theoretical Analysis and Applications in Data Storage
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A Combinatorial Methodology for Optimizing Non-Binary Graph-Based Codes: Theoretical Analysis and Applications in Data Storage

机译：优化基于非二进制图的代码的组合方法：理论分析和数据存储中的应用

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Non-binary (NB) low-density parity-check (LDPC) codes are graph-based codes that are increasingly being considered as a powerful error correction tool for modern dense storage devices. Optimizing NB-LDPC codes to overcome their error floor is one of the main code design challenges facing storage engineers upon deploying such codes in practice. Furthermore, the increasing levels of asymmetry incorporated by the channels underlying modern dense storage systems, e.g., multi-level Flash systems, exacerbate the error floor problem by widening the spectrum of problematic objects that contribute to the error floor of an NB-LDPC code. In a recent research, the weight consistency matrix (WCM) framework was introduced as an effective combinatorial NB-LDPC code optimization methodology that is suitable for modern Flash memory and magnetic recording (MR) systems. The WCM framework was used to optimize codes for asymmetric Flash channels, MR channels that have intrinsic memory, in addition to canonical symmetric additive white Gaussian noise channels. In this paper, we provide an in-depth theoretical analysis needed to understand and properly apply the WCM framework. We focus on general absorbing sets of type two (GASTs) as the detrimental objects of interest. In particular, we introduce a novel tree representation of a GAST called the unlabeled GAST tree, using which we prove that the WCM framework is optimal in the sense that it operates on the minimum number of matrices, which are the WCMs, to remove a GAST. Then, we enumerate WCMs and demonstrate the significance of the savings achieved by the WCM framework in the number of matrices processed to remove a GAST. Moreover, we provide a linear-algebraic analysis of the null spaces of WCMs associated with a GAST. We derive the minimum number of edge weight changes needed to remove a GAST via its WCMs, along with how to choose these changes. In addition, we propose a new set of problematic objects, namely oscillating sets of type two (OSTs), which contribute to the error floor of NB-LDPC codes with even column weights on asymmetric channels, and we show how to customize the WCM framework to remove OSTs. We also extend the domain of the WCM framework applications by demonstrating its benefits in optimizing column weight 5 codes, codes used over Flash channels with additional soft information, and spatially coupled codes. The performance gains achieved via the WCM framework range between 1 and nearly 2.5 orders of magnitude in the error floor region over interesting channels.
机译：非二进制（NB）低密度奇偶校验（LDPC）代码是基于图的代码，越来越多地被视为现代密集存储设备的强大纠错工具。优化NB-LDPC代码以克服其错误底限是在实际部署此类代码时存储工程师面临的主要代码设计挑战之一。此外，由现代密集存储系统（例如，多层闪存系统）下面的信道所包含的不对称程度的增加，通过扩大导致NB-LDPC码的错误基底的问题对象的范围而加剧了错误基底问题。在最近的研究中，引入了权重一致性矩阵（WCM）框架作为适用于现代闪存和磁记录（MR）系统的有效组合NB-LDPC代码优化方法。除了规范的对称加性白高斯噪声通道以外，WCM框架还用于优化非对称Flash通道，具有固有内存的MR通道的代码。在本文中，我们提供了深入的理论分析，以了解和正确应用WCM框架。我们专注于类型2的一般吸收集（GAST）作为感兴趣的有害对象。特别是，我们介绍了一种称为无标签GAST树的GAST的新颖树表示形式，使用该树表示，可以证明WCM框架是最佳的，因为它可以在最小数量的矩阵（即WCM）上操作以删除GAST 。然后，我们列举了WCM，并说明了WCM框架节省的成本在处理去除GAST的矩阵数量中的重要性。此外，我们提供了与GAST相关的WCM的零空间的线性代数分析。我们得出通过WCM移除GAST所需的最小边缘权重更改次数，以及如何选择这些更改。此外，我们提出了一组新的有问题的对象，即类型为2的振动集（OST），它们对非对称信道上具有均匀列权重的NB-LDPC码的错误基底作出了贡献，并且展示了如何自定义WCM框架删除OST。我们还将通过展示WCM框架在优化列权重5代码，具有附加软信息的Flash通道上使用的代码以及空间耦合代码方面的优势，扩展WCM框架应用程序的领域。在感兴趣的通道上，通过WCM框架在错误基底区域中获得的性能提升介于1到2.5个数量级之间。

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