Error floor analysis of left-regular LDPC codes

纠错码（ECC）是任何通信系统在存在噪声和干扰时保证通信可靠性的重要组成部分。包括低密度奇偶校验（LDPC）码在内的迭代编码方案是未来几代ECC的主要候选方案。LDPC码在光通信中应用的主要挑战之一是这种系统的错误率要求非常低（在$10^{-15}$到$10^{-17}$的数量级）。目前，在没有分析结果的情况下，即使在最快的计算机上，评估给定的代码并在如此低的错误率下可靠地估计其错误下限也需要几个月的蒙特卡罗模拟。硬件模拟也需要数周才能达到这样的错误率。存在各种各样的估计技术，主要基于重要性抽样理论，可以加快错误层估计的过程，但这些技术都依赖于关于代码中某些有问题的结构的可靠信息，这些结构被称为捕获集。毫不奇怪，获得这样的信息通常是一个非常困难的问题。因此，即使使用这种技术来更快地估计误差层，也不能保证估计是准确的。因此，文献中常见的做法是通过蒙特卡罗模拟来验证估计的准确性。这违背了设计比蒙特卡罗模拟更快的估计技术的全部目的。在这个项目中，目标是设计一种快速的技术，可以可靠地估计常规LDPC码的错误层。正则码在光通信中是特别感兴趣的，因为与它们的不规则对口物相比，它们已知具有较低的误差层。所提出的技术将使我们的工业合作伙伴Infinera能够有效地评估目前考虑在未来光学系统中可能采用的不同LDPC候选代码。这将为Infinera节省大量的时间和计算资源。它还为公司提供了一种分析工具，用于严格评估错误层区域的LDPC代码，这是目前行业所缺少的。

Error correcting codes (ECC) are an essential part of any communication system to ensure the reliability of communication at the presence of noise and interference. Iterative coding schemes including low-density parity-check (LDPC) codes are prime candidates for future generations of ECC. One of the main challenges for the application of LDPC codes in optical communications is the very low error rate requirement in such systems (in the oder of $10^{-15}$ to $10^{-17}$). Currently, in the absence of analytical results, evaluating a given code and reliably estimating its error floor at such low error rates would require months of Monte Carlo simulations even on the fastest computers. Hardware emulations would also take weeks to reach such error rates. There exist a variety of estimation techniques, mostly based on the theory of Importance Sampling, that can speed up the process of error floor estimation, but those techniques all rely on having reliable information about certain problematic structures in the code, referred to as trapping sets. Not surprisingly, attaining such information is in general a very hard problem. Threfore, even if one uses such techniques to estimate the error floor faster, there is no guarantee that the estimate is accurate. As a result, the common practice in the literature is to verify the accuracy of the estimates by Monte Carlo simulations. This defeats the whole purpose of devising estimation techniques that are faster than Monte Carlo simulations. In this project, the goal is to devise a fast technique that can reliably estimate the error floor of regular LDPC codes. Regular codes are of particular interest in optical communications as they are known to have lower error floors compared to their irregular counterparts. The proposed technique will allow our industrial partner, Infinera, to efficiently evaluate different LDPC code candidates currently considered for possible adoption in future optical systems. This will save Infinera a tremendous amount of time and computational resources. It also provides the company with an analytical tool for rigorous evaluation of LDPC codes in the error floor region, something that is currently missing in the industry.