Low-density parity-check codes together with belief propagation (BP) decoding are known to be well-performing for large block lengths. However, for short block lengths there is still a considerable gap between the performance of the BP decoder and the maximum likelihood decoder. Different ensemble decoding schemes such as, e.g., the automorphism ensemble decoder (AED), can reduce this gap in short block length regime. We propose a generalized AED (GAED) that uses automorphisms according to the definition in linear algebra. Here, an automorphism of a vector space is defined as a linear, bijective self-mapping, whereas in coding theory self-mappings that are scaled permutations are commonly used. We show that the more general definition leads to an explicit joint construction of codes and automorphisms, and significantly enlarges the search space for automorphisms of existing linear codes. Furthermore, we prove the concept that generalized automorphisms can indeed be used to improve decoding. Additionally, we propose a code construction of parity check codes enabling the construction of codes with suitably designed automorphisms. Finally, we analyze the decoding performances of the GAED for some of our constructed codes.

翻译：低密度奇偶校验码结合置信传播（BP）解码在大分组长度下表现优异，但在短分组长度下，BP解码器与最大似然解码器之间仍存在显著性能差距。不同集成解码方案（如自同构集成解码器（AED））可缩小短分组长度下的这一差距。我们提出一种广义AED（GAED），采用线性代数定义下的自同构。此处向量空间的自同构定义为线性双射自映射，而编码理论中常用缩放置换作为自映射。研究表明，这一定义可实现对码与自同构的显式联合构造，并显著扩展现有线性码自同构的搜索空间。此外，我们通过理论验证广义自同构可用于改进解码性能，并提出一种奇偶校验码的构造方法，使所构造的码具备恰当设计的自同构。最后，我们分析了部分构造码在GAED下的解码性能。