局部可修复码的线性规划界 II
Linear Programming Bounds for Locally Recovery Codes II
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中文总结 AI 辅助
提出一个多项式规模线性规划界，用于$q$元全符号局部可修复码，通过保留三块联合汉明重量而非单一距离，在多数测试案例中优于现有界，并精确确定多个最大码大小和线性维数。
中文摘要 AI 辅助
我们给出了一个多项式规模的线性规划界，用于具有局部性参数$(r,\delta)$的$q$元全符号局部可修复码，且不假设线性性。关键思想是，对于每一对有序码字和每一个选定的修复视图，保留辅助集、被修复坐标以及码的其余部分上的联合汉明重量——而不是像早期公式那样将此三元组折叠为单个距离。将这种三块分布在相同长度的修复视图上取平均，可得到将其与全局距离分布联系起来的精确恒等式，以及同时编码局部性和谱正性的非负乘积-克拉夫丘克约束。由此产生的线性规划具有多项式个变量，其最优值优于普通Delsarte界，并且早期的外部距离公式、Li--Wei--Xiong的凸包界以及Gruica--Jany--Ravagnani的基于对偶的界都作为更粗糙的边缘分布从其中导出。在$q=2,3,4$上的精确有理证书表明，在十五个测试案例中的十三个中，该界严格强于这些先前线性规划中的最佳者，确定了七个精确的最大码大小和十二个精确的最大线性维数。
英文摘要
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.
