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Impure codes exceeding the pure bounds for quantum local recovery

Carlos Galindo, Fernando Hernando, Helena Martín-Cruz, Ryutaroh Matsumoto

Abstract

Literature provides several bounds for quantum local recovery, which essentially consider the number of message qudits, the distance, the length, and the locality of the involved codes. We give a family of $J$-affine variety codes that result in impure CSS codes. These quantum codes exceed several of the above mentioned bounds that apply to pure quantum locally recoverable codes. We also discuss a connection between bounds on quantum local recovery and on weight-constrained stabilizer codes.

Impure codes exceeding the pure bounds for quantum local recovery

Abstract

Literature provides several bounds for quantum local recovery, which essentially consider the number of message qudits, the distance, the length, and the locality of the involved codes. We give a family of -affine variety codes that result in impure CSS codes. These quantum codes exceed several of the above mentioned bounds that apply to pure quantum locally recoverable codes. We also discuss a connection between bounds on quantum local recovery and on weight-constrained stabilizer codes.

Paper Structure

This paper contains 7 sections, 6 theorems, 56 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Classical locally recoverable codes, general quantum locally recoverable codes and their Singleton-like bounds
  4. Quantum locally recoverable codes from the CSS construction and their Singleton-like bounds
  5. Quantum codes from linear dual-containing codes with respect to the Hermitian inner product
  6. Impure quantum locally recoverable codes exceeding previous bounds
  7. Concluding Remarks

Key Result

Lemma 2

For every positive integer $m$, there exists a binary linear $[4m, 3m-1]_2$-code $\mathcal{C}$ with locality $3$ such that every parity check matrix $H$ for $\mathcal{C}$ always contains a row vector of Hamming weight at least $2m$. $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure 1: Points in $P_{H,V}$ with $H=5$, $V=3$ and $q=5$ in Example \ref{['ex1']}
  • Figure 2: Monomials in $\Delta_{H,V,a,b}$ with $H=5$, $V=3$ and $a=b=0$ in Example \ref{['ex1']}
  • Figure 3: Monomials in $\Delta_{H,V,a,b}$ with $H=V=8$ and $a=b=1$ in Example \ref{['ex1e']}

Theorems & Definitions (16)

  • Remark 1
  • Lemma 2
  • Remark 3
  • Example 4
  • Example 5
  • Proposition 6
  • Remark 7
  • Proposition 8
  • Theorem 9
  • Remark 10
  • ...and 6 more