Iterative Decoding Techniques and their Applications

在过去的十年中，迭代译码方法由于首先由Turbo码和最近由低密度奇偶校验码(LDPC)获得的惊人的差错性能而受到极大的关注。然而，由于这些结果中的绝大多数要么是理论推导的，要么是通过计算机仿真得到的，因此在LDPC码的实现中仍然存在一些问题。本研究解决了其中的一些问题，并可分为三个主要方面：(1)LDPC码的代数构造：本研究基于代数和有限几何来研究LDPC码的构造。这些构造的动机是以非常具体和明确的方式描述代码族的可能性，这对于借助计算机生成的代码是不可能的。此外，可以确定码的最小汉明距离等重要参数的紧界(如果不是精确的值)。在评估需要非常低误码率的应用程序(如存储系统)的错误性能时，此功能仍然很有意义。此外，所构造的码的代数结构可以被用来执行简单的编码或译码，例如基于线性反馈移位寄存器的循环或准循环码的编码。(2)中短长度LDPC码的近最佳减少延迟迭代译码：尽管对于长LDPC码已经推导并获得了非常令人印象深刻的结果，但较短长度的码更有可能用于实际应用，例如无线通信系统。对于中等长度的码，仅迭代译码需要大量迭代，并且显著低于最佳差错性能。研究人员设计了新的译码方法，与迭代译码相结合，既减少了所需的迭代次数，又提高了现有方法的误码性能。(3)迭代译码方法的简化：LDPC码的最佳迭代译码方法取决于操作信噪比，并且受到概率或对数似然比计算固有的数值问题的影响。研究人员研究了新的译码方法，这些方法变得普遍，对数值问题不那么敏感，同时保持了接近最佳的误码性能。这些特性对于在商业产品中实现这些有前景的新的解码方法是重要的。

Over the past decade, iterative decoding methods have received a great deal of interest due to the astonishing error performances achieved first by turbo codes, and more recently by low-density parity check (LDPC) codes. However, since the vast majority of these results have been either theoretically derived for, or achieved by computer simulations of very long codes, severalissues remain problematic in the implementation of LDPC codes.This research addresses some of these problems and can be divided into three major areas:(1) Algebraic construction of LDPC codes:This research investigates the construction of LDPC codes based on algebra and finite geometries. These constructions are motivated by the possibility of describing families of codes in a very specific and unambiguous way, which is impossible for codes generated with the aid of a computer. Furthermore, tight bounds if not exact values of important parameters such as the code minimum Hamming distance can be determined. This feature remains interesting to evaluate the error performance of applications requiring very small bit error rates such as storage systems. In addition, the algebraic structure of the codes constructed can be exploited to perform simple encoding or decoding, such as encoding of cyclic or quasi-cyclic codes based on linear feedback shift registers.(2) Near-optimum reduced-latency iterative decoding of short to medium length LDPC codes:Although very impressive results have been derived and achieved with long LDPC codes,codes of shorter lengths are more likely to be used in practical applications such as wireless communication systems. For codes of medium length, iterative decoding alone requires a lot of iterations and falls significantly short of the optimum error performance. The investigators devise new decoding methods which, combined with iterative decoding, both reduce the number ofrequired iterations and improve the error performance of current approaches.(3) Simplification of iterative decoding methods:The best iterative decoding methods for LDPC codes depend on the operating signal-to-noise ratio and are subject to numerical problems inherent to the evaluation of probabilities or log-likelihood ratios. The investigators study new decoding methods which become universal and aremuch less sensitive to numerical problems while maintaining near-optimum error performance. Such properties are important for implementing these promising new decoding approaches in commercial products.