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）译码在长码长条件下表现优异。然而在短码长条件下，BP译码器与最大似然译码器之间仍存在显著性能差距。诸如自同构集成译码器（AED）等不同的集成译码方案可缩短该差距。本文提出广义自同构集成译码器（GAED），其采用线性代数定义下的自同构概念。此处向量空间的自同构定义为线性双射自映射，而编码理论中通常使用缩放置换自映射。研究表明：该广义定义可显式联合构造码与自同构，并显著拓展现有线性码的自同构搜索空间。此外，我们通过理论证明广义自同构确实可用于改进译码性能，并提出一种奇偶校验码的构造方法，使得码字具有适宜设计的自同构结构。最后，对部分构造码的GAED译码性能进行了分析。