Alphabet-Dependent Bounds for Pure Quantum $(r,\rho)$-Locally Recoverable Codes

Abstract:A quantum $(r,\rho)$-locally recoverable code ($(r,\rho)$-qLRC) is a quantum code in which every qudit can be recovered from at most $r+\rho-1$ other qudits, even after $\rho-1$ 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 $(r,\rho)$-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.

Quantum Physics (quant-ph); Information Theory (cs.IT)

arXiv:2608.28650 [quant-ph]

(or arXiv:2608.28650v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2608.28650 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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