Multi-View Block Distance Distributions and Linear Programming Bounds for Locally Recoverable Codes with Availability

2026-09-30

AI Summary

This study addresses the limitation of existing single-view models in deriving tight linear programming (LP) bounds for locally recoverable codes with arbitrary fixed availability, where information loss inherently degrades bound quality. To overcome this, we propose a novel multi-view (s=2) four-block LP framework that preserves coupling relationships among repair sets by recording Hamming distance distributions across all partitions. By integrating central counting identities, product Krawtchouk positivity, and rational primal-dual certificates, our approach enables polynomial-size relaxation computations for nonlinear codes over finite fields. We explicitly characterize the information loss induced by coarsening and prove that the proposed multi-view bound is strictly tighter than both single-view and globally conditioned relaxations. Furthermore, we exactly determine several parameter values, including M_max(2,8,4,2,2,2)=8, confirming the optimality of our model across all examined instances.

Abstract

We develop a multi-view linear-programming framework for locally recoverable codes with arbitrary fixed availability $a$. For any retained order $1 \le s \le a$, 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.

Problem

• Locally Recoverable Codes
• Availability
• Linear Programming Bounds
• Distance Distributions
• Maximum Code Size

Innovation

• Locally Recoverable Codes
• Multi-View Linear Programming
• Availability
• Krawtchouk Positivity
• Distance Distributions