We establish a bijection between binary even LCD codes and simple graphs whose adjacency matrices are idempotent over $\FF_2$. This bijection preserves equivalence: permutation equivalence of codes corresponds exactly to graph isomorphism. Based on this framework, we characterize which distance-regular graphs yield LCD codes via intersection array parameters, prove that non-isomorphic conference graphs yield inequivalent codes, and classify LCD-derived graphs of small orders.

翻译：我们在二元偶LCD码与简单图之间建立了一一对应关系，其中简单图的邻接矩阵在$\FF_2$上满足幂等性。该对应关系保持等价性：码的置换等价性精确对应于图的同构。基于此框架，我们通过交集数组参数刻画了哪些距离正则图能产生LCD码，证明了非同构的会议图产生不等价的码，并对小阶数的LCD导出图进行了分类。