On optimal quantum LRCs from the Hermitian construction and $t$-designs

TL;DR

This paper advances the theory and construction of quantum locally recoverable codes (qLRCs) using Hermitian construction and t-designs, providing new bounds, infinite families of NMDS codes, and explicit optimal qLRCs with flexible parameters.

Contribution

It introduces four bounds for qLRCs, constructs new NMDS code families supporting t-designs, and derives explicit optimal qLRCs surpassing previous constructions.

Findings

Four bounds for qLRCs with asymptotic comparisons

New infinite families of NMDS codes supporting t-designs

Explicit optimal qLRCs with flexible parameters

Abstract

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 $et~al.$ (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 $t\in \{2,3\}$, 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.