Generalized Automorphisms of Channel Codes: Properties, Code Design, and a Decoder

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 BP decoding and maximum likelihood decoding. Different ensemble decoding schemes such as, e.g., automorphism ensemble decoding (AED), can reduce this gap in short block length regime. We propose 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 linear codes enabling the construction of codes with suitably designed automorphisms. Finally, we analyze the decoding performances of GAED for some of our constructed codes.