Linear Programming Bounds for Locally Recovery Codes II

2026-09-12

AI Summary

本文提出了一种多项式规模的线性规划方法，用于计算具有局部恢复参数(r,δ)的q进制全符号局部可恢复码的界限，无需假设线性。

Abstract

We give a polynomial-size linear programming bound for $q$-ary all-symbol locally recoverable codes with locality parameters $(r,δ)$, without assuming linearity. The key idea is to keep, for every ordered pair of codewords and every selected recovery view, the joint Hamming weight on the helper set, the recovered coordinate, and the rest of the code -- rather than collapsing this triple into a single distance, as earlier formulations do. Averaging this three-block distribution over recovery views of the same length yields exact identities linking it to the global distance distribution, together with nonnegative product-Krawtchouk constraints that encode locality and spectral positivity simultaneously. The resulting LP has polynomially many variables, its optimum dominates the ordinary Delsarte bound, and an earlier outside-distance formulation, the convex-hull bound of Li--Wei--Xiong, and the dual-based bound of Gruica--Jany--Ravagnani all arise from it as coarser marginals. Exact rational certificates over $q=2,3,4$ show the bound is strictly stronger than the best of these prior LPs in thirteen of fifteen tested cases, pinning down seven exact maximum code sizes and twelve exact maximum linear dimensions.

Problem

• linear programming
• locally recoverable codes
• Hamming weight
• locality parameters
• polynomial-size

Innovation

• polynomial-size linear programming
• joint Hamming weight
• locality parameters
• nonnegative product-Krawtchouk constraints
• global distance distribution