AI 中文总结
本文研究有限域上线性互补对偶（LCD）码的邻接关系及其诱导图，确定LCD码的LCD邻居数量，证明邻接图在任意有限域上的正则性及二元和奇特征情形下结构子图的正则性，为LCD码提供图论框架。
AI 中文摘要
线性互补对偶（LCD）码是一类重要的线性码，在密码学、经典纠错和量子编码理论中具有应用。本文研究了有限域上LCD码的邻接关系以及由该关系导出的图，其中两个码当且仅当它们的交集余维数为1时相邻。我们确定了LCD码的邻居中仍为LCD码的个数，并利用该结果分析了相应邻接图的结构。特别地，我们证明了该图在任意有限域上的正则性，并在二元和奇特征情形下建立了其主要结构子图的进一步正则性性质。这些结果为LCD码的研究提供了图论框架，并揭示了其邻域结构中强烈的组合正则性。
英文摘要
Linear complementary dual (LCD) codes form an important class of linear codes with applications in cryptography, classical error correction, and quantum coding theory. In this paper, we study the neighbor relation on LCD codes over finite fields and the graph induced by this relation, where two codes are adjacent whenever they intersect in codimension one. We determine the number of neighbors of an LCD code that are also LCD, and we use this result to analyze the structure of the corresponding neighbor graph. In particular, we prove its regularity over arbitrary finite fields and establish further regularity properties for its main structural subgraphs in the binary and odd-characteristic cases. These results provide a graph-theoretic framework for the study of LCD codes and reveal a strong combinatorial regularity in their neighborhood structure.
