On automorphism groups of binary cyclic codes

The article delves into the automorphism groups of binary cyclic codes, a pivotal area in coding theory where group actions on codes offer valuable insights into their structure. The study begins with an introduction to the significance of code automorphisms and reviews previous work on specific code classes like Hamming, Reed-Muller, and BCH codes. The main focus is on binary cyclic codes, for which the authors employ matrix presentation methods to characterize relatively large subgroups of automorphisms. The paper is organized into sections that cover the classification of full automorphism groups for cyclic codes of lengths 2p and with irreducible generator polynomials, where p is an odd prime. Notably, the authors introduce a new permutation group decoder based on automorphism groups and discuss the automorphism group of Polar codes. The paper concludes with open problems for future research, inviting further exploration into the classification of automorphisms for cyclic codes of various lengths.