On the Automorphism Groups of Berman Codes and associated Abelian Codes
2026-05-26 • Information Theory
Information Theory
AI summaryⓘ
The authors study special types of codes called Berman codes, which include Reed-Muller codes and are useful for communication over noisy channels. They focus on the automorphism group, which is the set of all ways to rearrange code bits without changing the code. The authors find the exact automorphism groups for Berman codes, their duals, and related abelian codes when a parameter n is at least 5. For the case when n equals 3, they provide partial and some complete descriptions of these groups.
Berman codesautomorphism groupbinary linear codesdual codesabelian codesReed-Muller codesbinary erasure channelAWGN channelcode automorphismscode decoding
Authors
Harshvardhan Pandey, Prasad Krishnan
Abstract
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.