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PreprintCC BY 4.0arXiv

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

  • Vijay Kumar
  • Ramakrishna Bandi
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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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Cite

BibTeXRIS
BibTeX
@misc{kumar2026alphabetdependent,
  title = {{Alphabet-Dependent Bounds for Pure Quantum $(r,ρ)$-Locally Recoverable Codes}},
  author = {Vijay Kumar and Ramakrishna Bandi},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.28650},
  eprint = {2608.28650},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.28650},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Alphabet-Dependent Bounds for Pure Quantum (r,ρ)-Locally Recoverable Codes
AU  - Vijay Kumar
AU  - Ramakrishna Bandi
PY  - 2026
DA  - 2026/08/19
PB  - arXiv
DO  - 10.48550/arxiv.2608.28650
UR  - https://arxiv.org/abs/2608.28650
AB  - 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.
ER  -