Quantum Locally Recoverable Codes via Good Polynomials

TL;DR

This paper introduces a new method for constructing quantum locally recoverable codes (qLRCs) using good polynomials, broadening the class of such codes and providing improved bounds on their minimum distance.

Contribution

It generalizes qLRC construction methods to include any good polynomial and develops new bounds on minimum distance without prime restrictions.

Findings

New qLRC construction method using any good polynomial.

A novel approach for designing good polynomials via affine groups.

Lower bounds on minimum distance without prime restrictions.

Abstract

Locally recoverable codes (LRCs) with locality parameter $r$ can recover any erased code symbol by accessing $r$ other code symbols. This local recovery property is of great interest in large-scale distributed classical data storage systems as it leads to efficient repair of failed nodes. A well-known class of optimal (classical) LRCs are subcodes of Reed-Solomon codes constructed using a special type of polynomials called good polynomials. Recently, Golowich and Guruswami initiated the study of quantum LRCs (qLRCs), which could have applications in quantum data storage systems of the future. The authors presented a qLRC construction based on good polynomials arising out of subgroups of the multiplicative group of finite fields. In this paper, we present a qLRC construction method that can employ any good polynomial. We also propose a new approach for designing good polynomials using subgroups of affine general linear groups. Golowich and Guruswami also derived a lower bound on the minimum distance of their qLRC under the restriction that $r+1$ is prime. Using similar techniques in conjunction with the expander mixing lemma, we develop minimum distance lower bounds for our qLRCs without the $r+1$ prime restriction.