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

• URL: http://arxiv.org/abs/2608.28650v1
• Date: Wed, 19 Aug 2026 05:31:21 GMT
• Title: Alphabet-Dependent Bounds for Pure Quantum $(r,ρ)$-Locally Recoverable Codes
• Abstract summary: A quantum $(r,)$-locally recoverable code ($(r,)$-qLRC) is a quantum code in which everydit can be recovered from at most $r+-1$ other qudits.<n>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.
• Score: 5.782427435542872
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: A quantum $(r,ρ)$-locally recoverable code ($(r,ρ)$-qLRC) is a quantum code in which every qudit can be recovered from at most $r+ρ-1$ other qudits, even after $ρ-1$ 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 $(r,ρ)$-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.

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