On Quantum Stabilizer Codes derived from Local Frobenius Rings

TL;DR

This paper investigates quantum stabilizer codes over local Frobenius rings, analyzing their minimum distances and isometries, and conjectures that free codes over rings perform at least as well as those over fields.

Contribution

It introduces the study of stabilizer codes over local Frobenius rings, compares their properties to those over fields, and explores their isometries with open problems.

Findings

Free stabilizer codes over rings often match or outperform those over fields.

The relative minimum distances of codes over rings are comparable to their reductions over residue fields.

Preliminary results on code isometries highlight open research directions.

Abstract

In this paper we consider stabilizer codes over local Frobenius rings. First, we study the relative minimum distances of a stabilizer code and its reduction onto the residue field. We show that for various scenarios, a free stabilizer code over the ring does not underperform the according stabilizer code over the field. This leads us to conjecture that the same is true for all free stabilizer codes. Secondly, we focus on the isometries of stabilizer codes. We present some preliminary results and introduce some interesting open problems.