Abelian Noncyclic Orbit Codes and Multishot Subspace Codes

TL;DR

This paper characterizes orbit codes as geometrically uniform, introduces Abelian non-cyclic orbit codes, and demonstrates their application to multishot subspace codes, reducing computational complexity in analyzing their minimum subspace distance.

Contribution

It provides a new geometric characterization of orbit codes, introduces Abelian non-cyclic orbit codes, and applies these to multishot subspace coding with reduced computational effort.

Findings

Geometric characterization of orbit codes as geometrically uniform.

Construction method for Abelian non-cyclic orbit codes.

Reduction in computations for minimum subspace distance.

Abstract

In this paper we characterize the orbit codes as geometrically uniform codes. This characterization is based on the description of all isometries over a projective geometry. In addition, the Abelian orbit codes are defined and a new construction of Abelian non-cyclic orbit codes is presented. In order to analyze their structures, the concept of geometrically uniform partitions have to be reinterpreted. As a consequence, a substantial reduction in the number of computations needed to obtain the minimum subspace distance of these codes is achieved and established. An application of orbit codes to multishot subspace codes obtained according to a multi-level construction is provided.