多视图块距离分布与具有可用性的局部可恢复码的线性规划界

中文摘要 AI 辅助

我们为具有任意固定可用性 $a$ 的局部可恢复码开发了一个多视图线性规划框架。对于任意保留阶数 $1 \le s \le a$，所选的辅助集、被恢复的坐标及其补集构成一个 $(s+2)$ 部分划分。记录所有块上的汉明距离既保留了所选修复备选方案之间的兼容性，也保留了它们与剩余坐标的耦合。所得的联合分布满足中心计数恒等式、乘积-克拉夫丘克正性和来自局部距离条件的碰撞不等式；对于固定的 $s$，这些约束为任意有限域上的任意（可能非线性的）码给出了一个多项式规模的松弛。保留一个视图可恢复我们伴随论文中的三块模型。我们详细发展了第一个真正的多视图情形 $s=2$，并将精确计算特化到可用性 $a=2$。所得的四块线性规划投影到单视图三块模型和全局条件化的双视图松弛，明确展示了每次粗化所丢失的信息。精确的有理数原始-对偶证书以及经过验证的构造证明了 $M_{\max}(2,8,4,2,2,2)=8$，$M_{\max}(3,7,3,2,2,2)=27$，$M_{\max}(4,7,3,2,2,2)=64$；在所有三种情况下，四块界都严格强于两种粗化。

英文摘要

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.

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16 pages, 2 tables