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Quantum Locally Recoverable Codes via Good Polynomials

Sandeep Sharma, Vinayak Ramkumar, Itzhak Tamo

TL;DR

A qLRC construction method that can employ any good polynomials arising out of subgroups of the multiplicative group of finite fields is presented and a new approach for designing good polynomials using subgroups of affine general linear groups is proposed.

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.

Quantum Locally Recoverable Codes via Good Polynomials

TL;DR

A qLRC construction method that can employ any good polynomials arising out of subgroups of the multiplicative group of finite fields is presented and a new approach for designing good polynomials using subgroups of affine general linear groups is proposed.

Abstract

Locally recoverable codes (LRCs) with locality parameter can recover any erased code symbol by accessing 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 is prime. Using similar techniques in conjunction with the expander mixing lemma, we develop minimum distance lower bounds for our qLRCs without the prime restriction.

Paper Structure

This paper contains 15 sections, 15 theorems, 60 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Classical Locally Recoverable Codes
  4. Quantum Locally Recoverable codes
  5. Prior Work on qLRCs
  6. Constructions
  7. Good Polynomials from AGL
  8. Group Action
  9. Affine General Linear Group
  10. Good polynomials via subgroups of AGL
  11. Distance Bounds
  12. Conclusions and future work
  13. Proof of Lemma \ref{['lem:good_poly']}
  14. Proof of Lemma \ref{['calc']}
  15. Proof of Lemma \ref{['lem:mult_add']}

Key Result

Proposition 3

Let $\mathcal{C}_i$ be an $[n,k_i]_q$ linear code for $i=1,2.$ If $\mathcal{C}_1^{\perp}\subseteq \mathcal{C}_2$, then there exists an $[[n,\kappa,\delta]]_q$-quantum code $\mathcal{C}=\text{CSS}(\mathcal{C}_1,\mathcal{C}_2)$ with $\kappa=k_1+k_2-n$ and $\delta=\min\{\text{wt}(\mathcal{C}_1\setminus

Figures (1)

  • Figure 1: Comparison between various bounds for $n=63, r=6, q=64$. Here GG bound is the bound in Golowich, our bound is the bound in \ref{['eq:distance_prime']}, and degree bound is the bound in \ref{['eq:bound_degree']}.

Theorems & Definitions (26)

  • Definition 1: Locally Recoverable Code
  • Definition 2: Quantum Code
  • Proposition 3: CSS Construction Ketkar
  • Definition 4: Quantum Locally Recoverable Code Golowich
  • Proposition 5: Golowich
  • Corollary 6
  • Definition 7: Tamo
  • Lemma 9
  • Theorem 10
  • proof
  • ...and 16 more