On the Automorphism Groups of Berman Codes and associated Abelian Codes
For coding theorists, this provides exact automorphism group characterizations for a family of codes with practical relevance, though the results are incremental extensions of known algebraic structures.
The paper identifies the exact automorphism groups for Berman codes and their duals, and for a class of related abelian codes when n≥5, with partial results for n=3. This extends prior work on code automorphisms and builds on recent capacity-achieving results for Berman codes.
The automorphism group of a code is the group of permutations that map a code to itself. Berman codes are a class of binary linear codes characterized by two integer parameters $n\geq 2$ and $m\geq 1$, and this class includes the Reed-Muller codes as well. The class of Berman codes and their duals were recently shown to achieve the capacity of the binary erasure channel. A number of abelian codes that arise from the intersection and subspace sums of Berman and Dual Berman codes were also identified recently, for odd $n\geq 3$. A subclass of these abelian codes was shown to have good short block-length performance for AWGN channels, with efficient decoding algorithms. In this work, we identify the exact automorphism group for Berman codes and their duals. Further, we find the exact automorphism group for the above mentioned abelian codes, when $n\geq 5$. In the case of such abelian codes with $n=3$, we present partial characterizations of the automorphism groups for a large collection of parameter choices, and complete characterizations for a few.