Constructions of Subsystem Codes over Finite Fields

TL;DR

This paper presents new methods for constructing quantum subsystem codes over finite fields, including derivations from classical cyclic codes and techniques to optimize code parameters, resulting in many optimal codes.

Contribution

It introduces novel constructions and methods for deriving, extending, and optimizing quantum subsystem codes from classical codes, enhancing their versatility and performance.

Findings

Many optimal subsystem codes are obtained.

Methods to trade dimensions while maintaining minimum distance.

Tables of bounds on subsystem code parameters are provided.

Abstract

Subsystem codes protect quantum information by encoding it in a tensor factor of a subspace of the physical state space. Subsystem codes generalize all major quantum error protection schemes, and therefore are especially versatile. This paper introduces numerous constructions of subsystem codes. It is shown how one can derive subsystem codes from classical cyclic codes. Methods to trade the dimensions of subsystem and co-subystem are introduced that maintain or improve the minimum distance. As a consequence, many optimal subsystem codes are obtained. Furthermore, it is shown how given subsystem codes can be extended, shortened, or combined to yield new subsystem codes. These subsystem code constructions are used to derive tables of upper and lower bounds on the subsystem code parameters.