The greatest-common-divisor (GCD) method is a general framework employing a set of simple inequalities (called GCD constraint) to guarantee girth eight for a class of $(J,L)$ quasi-cyclic (QC) low-density parity-check (LDPC) codes. However, an important problem, i.e., whether the GCD constraint is necessary for this class of codes to have girth eight, remains open. In this letter, the question is answered affirmatively, following which a novel algorithm aiming to find the shortest codes with girth eight in such a class is proposed. Besides, a close connection is established between the GCD method and the base expansion method, which are both applicable for any $J$ and any $L$ .

中文翻译：

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GCD 约束与全长行乘法器 QC-LDPC 码之间的关系，带环八

最大公约数 (GCD) 方法是一个通用框架，它采用一组简单的不等式（称为 GCD 约束）来保证一类的周长为 8 $(J,L)$ 准循环 (QC) 低密度奇偶校验 (LDPC) 码。然而，一个重要的问题，即 GCD 约束是否对这类代码具有 8 周长是必要的，仍然是开放的。在这封信中，这个问题得到了肯定的回答，随后提出了一种新算法，旨在寻找此类中周长为 8 的最短代码。此外，GCD 方法和基扩展方法之间建立了紧密的联系，这两种方法都适用于任何 $J$ 和任何 $L$ .