On the Automorphism Groups and Equivalence of Cyclic Combinatorial Objects

TL;DR

This paper classifies automorphism groups of cyclic combinatorial objects, including cyclic codes, and provides criteria for their equivalence, advancing understanding of their symmetry structures.

Contribution

It uniquely determines automorphism groups of cyclic combinatorial objects and characterizes permutations that establish their equivalence.

Findings

Automorphism groups of cyclic combinatorial objects are classified.

Automorphism groups of cyclic codes are specifically characterized.

Criteria for equivalence of cyclic combinatorial objects are established.

Abstract

We determine the permutation groups that arise as the automorphism groups of cyclic combinatorial objects. As special cases we classify the automorphism groups of cyclic codes. We also give the permutations by which two cyclic combinatorial objects on $p^m$ elements are equivalent.