Bounds for General (r,δ)-Quantum Locally Recoverable Codes
Abstract
We study general (r,δ)-quantum locally recoverable codes (qLRCs) without assuming purity or a stabilizer structure, and with arbitrarily overlapping recovery blocks. Using a selection lemma for multiple-erasure recovery, we establish a Singleton-like upper bound for general quantum subspaces. When δ=2, this bound reduces to the general single-erasure bound of Golowich and Guruswami. An integer optimization yields an explicit remainder correction that tightens the bound by up to δ−2. For pure codes, local decoupling and entropy arguments extend the parameter bound of Galindo et al., originally obtained for pure stabilizer codes constructed from Euclidean or Hermitian dual-containing classical codes, to arbitrary pure quantum subspaces. We also remove the partition assumption from several existing pure-code bounds for disjoint recovery blocks and obtain a further remainder correction by optimizing the budget. Finally, we analyze three families attaining these bounds: pure codes with overlapping recovery blocks and d=δ, which exhibit both a strict improvement of the pure-code bound and simultaneous equality in the pure-code and general bounds; impure codes with disjoint recovery blocks attaining the general bound; and an existing family of pure codes with overlapping recovery blocks and d>δ, which simultaneously attains all these bounds.
Related
- Alphabet-Dependent Bounds for Pure Quantum (r,ρ)-Locally Recoverable Codes
- Asymmetric Entanglement-Assisted Quantum Locally Recoverable Codes
- Asymptotically Good Quantum Locally Testable Codes
- Good Quantum Locally Testable Codes from Product Expansion
- NLTM Hamiltonians from gauged sheaf quantum locally testable codes
BibTeX
@misc{ma2026bounds,
title = {{Bounds for General $(r,δ)$-Quantum Locally Recoverable Codes}},
author = {Shuowei Ma and Fei Shi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40273},
eprint = {2610.40273},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40273},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Bounds for General (r,δ)-Quantum Locally Recoverable Codes
AU - Shuowei Ma
AU - Fei Shi
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40273
UR - https://arxiv.org/abs/2610.40273
AB - We study general (r,δ)-quantum locally recoverable codes (qLRCs) without assuming purity or a stabilizer structure, and with arbitrarily overlapping recovery blocks. Using a selection lemma for multiple-erasure recovery, we establish a Singleton-like upper bound for general quantum subspaces. When δ=2, this bound reduces to the general single-erasure bound of Golowich and Guruswami. An integer optimization yields an explicit remainder correction that tightens the bound by up to δ−2. For pure codes, local decoupling and entropy arguments extend the parameter bound of Galindo et al., originally obtained for pure stabilizer codes constructed from Euclidean or Hermitian dual-containing classical codes, to arbitrary pure quantum subspaces. We also remove the partition assumption from several existing pure-code bounds for disjoint recovery blocks and obtain a further remainder correction by optimizing the budget. Finally, we analyze three families attaining these bounds: pure codes with overlapping recovery blocks and d=δ, which exhibit both a strict improvement of the pure-code bound and simultaneous equality in the pure-code and general bounds; impure codes with disjoint recovery blocks attaining the general bound; and an existing family of pure codes with overlapping recovery blocks and d>δ, which simultaneously attains all these bounds.
ER -