  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.
