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arXiv:2608.10912

CSS Quantum LRCs with Intersecting Recovery Sets: Constructions and Bounds

Evagoras Stylianou, Vinayak Ramkumar, Holger Boche, Rawad Bitar

cs.ITquant-phSubmitted Aug 11, 2026• Announced Aug 12, 2026• 0 citations
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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.

Comments
Accepted for presentation at the 2026 IEEE Information Theory Workshop (ITW 2026)

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In this work, we study (r,t,x)(r,t,x)(r,t,x) quantum locally recoverable codes (qLRCs) with locality rrr, ttt recovery sets per qudit, and intersection parameter xxx. We first show that, assuming the underlying classical codes have dual minimum distance at least two, a CSS code is an (r,t,x)(r,t,x)(r,t,x)-qLRC if and only if the underlying classical codes are (r,t,x)(r,t,x)(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)(r,t,x)(r,t,x)-cLRCs, which yield binary (r,t,x)(r,t,x)(r,t,x)-qLRCs via the CSS construction. For CSS (r,t,x)(r,t,x)(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.
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In a recent work, quantum locally recoverable codes (qLRCs) have been introduced for their potential application in large-scale quantum data storage and implication for quantum LDPC codes. This work focuses on the bounds and constructions of qLRCs derived from the Hermitian construction, which solves an open problem proposed by Luo et al.et~al.et al. (IEEE Trans. Inf. Theory, 71 (3): 1794-1802, 2025). We present four bounds for qLRCs and give comparisons in terms of their asymptotic formulas. We construct several new infinite families of NMDS codes, with general and flexible dimensions, that support t-designs for t∈{2,3}t\in \{2,3\}t∈{2,3}, and apply them to obtain Hermitian dual-containing classical LRCs (cLRCs). As a result, we derive three explicit families of optimal qLRCs. Compared to the known qLRCs obtained by the CSS construction, our optimal qLRCs offer new and more flexible parameters. It is also worth noting that the constructed cLRCs themselves are interesting as they are optimal with respect to four distinct bounds for cLRCs.
Constructions of Quantum (r,δ)(r,δ)(r,δ)-LRCs from cyclic codes
Classical (r,δ)(r,δ)(r,δ) locally recoverable codes (LRCs) play a central role in distributed data storage systems as they enable an efficient recovery from erasures by accessing a small number of surviving symbols. Motivated by their prospective use in future quantum data storage and by recent theoretical progress on quantum locally recoverable codes (qLRCs), we investigate the construction of qLRCs from classical cyclic (r,δ)(r,δ)(r,δ)-LRCs. Our approach identifies cyclic LRCs whose defining sets satisfy a dual-containing condition, allowing them to serve as valid CSS ingredients. We present three explicit families of (r,δ)(r,δ)(r,δ)-qLRCs, two of which are optimal with respect to the quantum Singleton-like bound, whenever the codes are pure, thereby providing optimal examples. Additionally, the codes presented in Constructions 2 and 3 have no bound on their lengths with respect to the field size required to obtain these codes.
Alphabet-Dependent Bounds for Pure Quantum (r,ρ)(r,ρ)(r,ρ)-Locally Recoverable Codes
A quantum (r,ρ)(r,ρ)(r,ρ)-locally recoverable code ((r,ρ)(r,ρ)(r,ρ)-qLRC) is a quantum code in which every qudit can be recovered from at most r+ρ−1r+ρ-1r+ρ−1 other qudits, even after ρ−1ρ-1ρ−1 additional erasures inside the recovery set. The bounds currently known for this class, namely the Singleton-like and the GG Singleton-like bounds, are alphabet independent and are therefore loose for small-to-moderate qudit dimensions. In this letter, we derive three alphabet-dependent upper bounds for pure (r,ρ)(r,ρ)(r,ρ)-qLRCs obtained through the Hermitian CSS construction: a Griesmer-like, a Plotkin-like, and a sphere-packing-like bound. We further establish the asymptotic hierarchy among these bounds and identify the relative-distance regions in which each of them yields the tightest rate constraint.