Low-density parity-check (LDPC) codes have been the subject of much interest due to the fact that they can perform near the Shannon limit. In this paper we present a construction of LDPC codes from cubic symmetric graphs. The constructed codes are $(3,3)$-regular and the vast majority of the corresponding Tanner graphs have girth greater than four. We analyse properties of the obtained codes and present bounds for the code parameters, the dimension and the minimum distance. Furthermore, we give an expression for the variance of the syndrome weight of the constructed codes. Information on the LDPC codes constructed from bipartite cubic symmetric graphs with less than 200 vertices is presented as well. Some of the constructed codes are optimal, and some have an additional property of being self-orthogonal or linear codes with complementary dual (LCD codes).

中文翻译：

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由三次对称图构造的 LDPC 码

低密度奇偶校验 (LDPC) 码因其可以在香农极限附近执行的事实而备受关注。在本文中，我们提出了从三次对称图构建 LDPC 码的方法。构造的代码是 $(3,3)$-regular 并且绝大多数对应的 Tanner 图的周长大于 4。我们分析获得的代码的属性并给出代码参数、维度和最小距离的界限。此外，我们给出了构造代码的综合症权重方差的表达式。还提供了有关由少于 200 个顶点的二部三次对称图构造的 LDPC 码的信息。一些构造的代码是最优的，