Automorphism Groups and Isometries for Cyclic Orbit Codes

TL;DR

This paper investigates the automorphism groups and isometries of cyclic orbit codes in finite field extensions, providing structural insights and characterizations for these codes and their symmetries.

Contribution

It establishes the containment of automorphism groups within the normalizer of the Singer subgroup for certain orbit codes and characterizes linear isometries between these codes.

Findings

Automorphism group is contained in the normalizer of the Singer subgroup under specific conditions.

Generalization to orbits under the normalizer of the Singer subgroup with some exceptional cases.

Characterization of linear isometries between cyclic orbit codes.

Abstract

We study orbit codes in the field extension ${\mathbb F}_{q^n}$. First we show that the automorphism group of a cyclic orbit code is contained in the normalizer of the Singer subgroup if the orbit is generated by a subspace that is not contained in a proper subfield of ${\mathbb F}_{q^n}$. We then generalize to orbits under the normalizer of the Singer subgroup. In that situation some exceptional cases arise and some open cases remain. Finally we characterize linear isometries between such codes.