Quasi-cyclic (QC) LDPC codes with large girths play a crucial role in several research and application fields, including channel coding, compressed sensing and distributed storage systems. A major challenge in respect of the code construction is how to obtain such codes with the shortest possible length (or equivalently, the smallest possible circulant size) using algebraic methods instead of search methods. The greatest-common-divisor (GCD) framework we previously proposed has algebraically constructed QC-LDPC codes with column weights of 5 and 6, very short lengths, and a girth of 8. By introducing the concept of a mirror sequence and adopting a new row-regrouping scheme, QC-LDPC codes with column weights of 7 and 8, very short lengths, and a girth of 8 are proposed for arbitrary row weights in this article via an algebraic manner under the GCD framework. Thanks to these novel algebraic methods, the lower bounds (for column weights 7 and 8) on consecutive circulant sizes are both improved by asymptotically about 20%, compared with the existing benchmarks. Furthermore, these new constructions can also offer circulant sizes asymptotically about 25% smaller than the novel bounds.

翻译：具有大围长的准循环（QC）LDPC码在信道编码、压缩感知和分布式存储系统等多个研究和应用领域中发挥着至关重要的作用。在码构造方面的一个主要挑战是如何使用代数方法（而非搜索方法）获得具有尽可能短长度（或等价地，尽可能小循环子矩阵尺寸）的此类码。我们先前提出的最大公约数（GCD）框架已通过代数方法构造出列重为5和6、长度极短且围长为8的QC-LDPC码。本文通过引入镜像序列的概念并采用新的行重组方案，在GCD框架下以代数方式为任意行重提出了列重为7和8、长度极短且围长为8的QC-LDPC码。得益于这些新颖的代数方法，相较于现有基准，连续循环子矩阵尺寸的下界（针对列重7和8）均渐进提升了约20%。此外，这些新构造还能提供比新下界渐进小约25%的循环子矩阵尺寸。