Quasi-optimal cyclic orbit codes

TL;DR

This paper investigates cyclic orbit codes, establishing invariants, bounds, and existence results for quasi-optimal codes, and analyzes their automorphism groups within finite field vector spaces.

Contribution

It introduces a connection between cyclic orbit codewords and linear sets, derives bounds, and proves the existence of quasi-optimal codes in even-dimensional spaces.

Findings

New bounds on code parameters

Existence theorem for quasi-optimal codes

Automorphism group descriptions

Abstract

We focus on two aspects of cyclic orbit codes: invariants under equivalence and quasi-optimality. Regarding the first aspect, we establish a connection between the codewords of a cyclic orbit code and a certain linear set on the projective line. This allows us to derive new bounds on the parameters of the code. In the second part, we study a particular family of (quasi-)optimal cyclic orbit codes and derive a general existence theorem for quasi-optimal codes in even-dimensional vector spaces over finite fields of any characteristic. Finally, for our particular code family we describe the automorphism groups under the general linear group and a suitable Galois group.