The automorphism groups of various linear codes are well-studied yielding valuable insights into the respective code structure. This knowledge is successfully applied in, e.g., theoretical analysis and in improving decoding performance motivating the analyses of endomorphisms of linear codes. In this work, we discuss the structure of the set of transformation matrices of code endomorphisms, defined as a generalization of code automorphisms, and provide an explicit construction of a bijective mapping between the image of an endomorphism and its canonical quotient space. Furthermore, we introduce a one-to-one mapping between the set of transformation matrices of endomorphisms and a larger linear block code enabling the use of well-known algorithms for the search for suitable endomorphisms. Additionally, we propose an approach to obtain unknown code endomorphisms based on automorphisms of the code. Furthermore, we consider ensemble decoding as a possible use case for endomorphisms by introducing endomorphism ensemble decoding. Interestingly, EED can improve decoding performance when other ensemble decoding schemes are not applicable.

翻译：各类线性码的自同构群已被深入研究，揭示了各自码结构的宝贵信息。该知识成功应用于理论分析及提升译码性能等领域，从而推动了线性码自同态分析的进展。本文探讨了码自同态变换矩阵集合的结构（该结构可视为码自同构的推广），并明确构造了自同态像与其典范商空间之间的双射映射。进一步，我们建立了自同态变换矩阵集合与更大线性分组码之间的一一映射，从而可利用经典算法搜索合适的自同态。此外，我们提出了一种基于码自同构获取未知码自同态的方法。同时，我们将自同态集译码（EED）作为自同态的一种潜在应用场景进行探讨。值得注意的是，当其他集译码方案不可用时，EED能够有效提升译码性能。