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An efficient exhaustive low-weight codeword search for structured LDPC codes

机译：针对结构化LDPC码的高效穷举低权码字搜索

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摘要

In this paper, we present an algorithm to find all low-weight codewords in a given quasi-cyclic (QC) low-density parity-check (LDPC) code with a fixed column-weight and girth. The main idea is to view a low-weight codeword as an (a, 0) trapping sets, and then show that each topologically different (a, 0) trapping set can be dissected into smaller trapping sets. The proposed search method relies on the knowledge of possible topologies of such smaller trapping sets present in a code ensemble, which enables an efficient search. Combined with the high-rate QC LDPC code construction which successively adds blocks of permutation matrices, the algorithm ensures that in the code construction procedure all codewords up to a certain weight are avoided, which leads to a code with the desired minimum distance. The algorithm can be also used to determine the multiplicity of the low-weight codewords with different trapping set structure.
机译：在本文中，我们提出了一种算法，该算法可在给定的准循环（QC）低密度奇偶校验（LDPC）码中找到固定权重和围长的所有低权重码字。主要思想是将低权重代码字视为（a，0）陷阱集，然后证明可以将每个拓扑不同（a，0）的陷阱集分解为较小的陷阱集。所提出的搜索方法依赖于代码集中存在的此类较小陷阱集的可能拓扑的知识，从而可以进行有效的搜索。该算法与高效率的QC LDPC码构造相结合，可依次添加置换矩阵块，从而确保在码构造过程中避免使用所有达到一定权重的码字，从而得到具有所需最小距离的码。该算法还可以用于确定具有不同陷印集结构的低权码字的多样性。

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