CSS Quantum LRCs with Intersecting Recovery Sets: Constructions and Bounds

2026-08-11

AI Summary

This work investigates CSS-type quantum locally recoverable codes with locality $ r $, $ t $ recovery sets per qubit, and pairwise recovery set intersections of size $ x $, denoted as $ (r,t,x) $-qLRCs. It establishes, for the first time, an equivalence between such quantum codes and classical dual-containing locally recoverable codes, and introduces a novel construction based on subset-inclusion matrices. Under the assumptions of pure states and exact recovery, the study derives several Singleton-type bounds and constructs families of binary $ (r,t,x) $-qLRCs achieving high code rates and nontrivial minimum distances. These constructions demonstrate both the tightness of the derived bounds and the efficacy of the proposed method.

Abstract

In this work, we study $(r,t,x)$ quantum locally recoverable codes (qLRCs) with locality $r$, $t$ recovery sets per qudit, and intersection parameter $x$. We first show that, assuming the underlying classical codes have dual minimum distance at least two, a CSS code is an $(r,t,x)$-qLRC if and only if the underlying classical codes are $(r,t,x)$ classical LRCs (cLRCs) with common recovery sets. We then use subset-inclusion matrices to construct families of binary dual-containing $(r,t,x)$-cLRCs, which yield binary $(r,t,x)$-qLRCs via the CSS construction. For CSS $(r,t,x)$-qLRCs, we derive upper bounds on the dimension and rate, minimum-distance bounds in the pure case, and a Singleton-like dimension bound in the exact case. Finally, we show that these families attain high rates and nontrivial minimum distances.

Problem

• quantum locally recoverable codes
• CSS codes
• recovery sets
• intersection parameter
• locality

Innovation

• quantum locally recoverable codes
• CSS construction
• subset-inclusion matrices
• recovery sets intersection
• classical-quantum code correspondence