On Small Automorphism Groups of Binary Self-Dual Codes

In this paper, we investigate the structure of small automorphism groups of binary self-dual codes and present new theoretical results. We prove that the automorphism group of a binary self-dual code cannot be generated by a single cycle permutation of length 2 or 3, and that no automorphism can act exclusively on one half of the code’s coordinates. Additionally, we provide new constraints for binary self-dual codes with trivial automorphism groups. These findings have significant implications for the classification and construction of such codes, particularly in cases where the existence of a self-dual code remains unresolved, for example, in the case of a binary self-dual (72,36,16)-code. Our results concerning such a code with automorphism group isomorphic to the cyclic group of order 5 offer valuable insight for both constructive approaches and exhaustive computational searches in this context.