Multi-View Block Distance Distributions and Linear Programming Bounds for Locally Recoverable Codes with Availability
Abstract
We develop a multi-view linear-programming framework for locally recoverable codes with arbitrary fixed availability a. For any retained order 1 ≤ s ≤ a, the selected helper sets, the recovered coordinate, and their complement form an (s+2)-part partition. Recording the Hamming distance on all blocks preserves both compatibility among the selected repair alternatives and their coupling with the remaining coordinates. The resulting joint distribution satisfies centered counting identities, product-Krawtchouk positivity, and collision inequalities from the local-distance condition; for fixed s, these constraints give a polynomial-size relaxation for arbitrary, possibly nonlinear, codes over any finite field. Retaining one view recovers the three-block model of our companion paper. We develop the first genuinely multi-view case, s=2, in detail and specialize the exact computations to availability a=2. The resulting four-block LP projects to both the one-view three-block model and a globally conditioned two-view relaxation, making explicit the information lost by each coarsening. Exact rational primal-dual certificates together with checked constructions prove Mmax(2,8,4,2,2,2)=8, Mmax(3,7,3,2,2,2)=27, and Mmax(4,7,3,2,2,2)=64; in all three cases the four-block bound is strictly stronger than both coarsenings.
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BibTeX
@misc{kang2026multiview,
title = {{Multi-View Block Distance Distributions and Linear Programming Bounds for Locally Recoverable Codes with Availability}},
author = {Ming-Hsuan Kang and Maosheng Xiong and Yu Hsuan Hsieh and Po-Wei Lai},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39230},
eprint = {2609.39230},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39230},
}
RIS
TY - UNPB
TI - Multi-View Block Distance Distributions and Linear Programming Bounds for Locally Recoverable Codes with Availability
AU - Ming-Hsuan Kang
AU - Maosheng Xiong
AU - Yu Hsuan Hsieh
AU - Po-Wei Lai
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39230
UR - https://arxiv.org/abs/2609.39230
AB - We develop a multi-view linear-programming framework for locally recoverable codes with arbitrary fixed availability a. For any retained order 1 ≤ s ≤ a, the selected helper sets, the recovered coordinate, and their complement form an (s+2)-part partition. Recording the Hamming distance on all blocks preserves both compatibility among the selected repair alternatives and their coupling with the remaining coordinates. The resulting joint distribution satisfies centered counting identities, product-Krawtchouk positivity, and collision inequalities from the local-distance condition; for fixed s, these constraints give a polynomial-size relaxation for arbitrary, possibly nonlinear, codes over any finite field. Retaining one view recovers the three-block model of our companion paper. We develop the first genuinely multi-view case, s=2, in detail and specialize the exact computations to availability a=2. The resulting four-block LP projects to both the one-view three-block model and a globally conditioned two-view relaxation, making explicit the information lost by each coarsening. Exact rational primal-dual certificates together with checked constructions prove M_max(2,8,4,2,2,2)=8, M_max(3,7,3,2,2,2)=27, and M_max(4,7,3,2,2,2)=64; in all three cases the four-block bound is strictly stronger than both coarsenings.
ER -