强化零错误香农容量的递归构造
Strengthening Recursive Constructions for Zero-Error Shannon Capacity
- School of Electrical, Computing and Software Engineering（电气、计算与软件工程学院）
- University of Arizona（亚利桑那大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对零错误信息论中奇环香农容量的开放问题，通过异构改进递归构造，提升了7-环的香农容量下界，阐明了递归构造的通用原理。
AI中文摘要：
除5-环$C_5$外，所有奇环的精确香农容量均未知，这使奇环成为零错误信息论的核心开放问题。改进已知下界需要在这些图的强幂中构造大独立集。近期AI辅助研究取得了一系列快速进展：Itty等人的构造基础上，Gao开发了一种用于组合结构化独立集的递归乘积构造，随后Buys、Polak和Zuiddam（BPZ）通过更丰富的递归框架对其进行了强化。我们延续这一AI辅助探索路线，引入这些构造的异构改进。核心观察是，中间构造的效用不仅取决于其当前主独立集的大小，还取决于其带入后续递归的辅助结构。因此，该辅助结构的不同部分无需使用相同的独立集，递归中的不同出现也无需使用相同的中间表示。我们对Gao的二元乘积将此形式化，推导明确的传播规则，表明异构选择如何在保持当前代码大小不变的同时强化所得构件，随后将该原理扩展到更通用的BPZ框架，针对递归中的不同角色定制构造。将这些改进应用于7-环$C_7$，我们在$C_7^{\boxtimes 500}$中获得一个独立集，使得$\Theta(C_7)\ge 3.25883262\ldots$，改进了已知的最佳下界。除数值增益外，结果还阐明了递归零错误构造的一般原理：具有相同维度和当前代码大小的中间结构，其下游价值可根据在递归中的使用位置和方式而不同。
英文摘要：
The exact Shannon capacity is unknown for every odd cycle beyond the five-cycle $C_5$, making odd cycles a central open problem in zero-error information theory. Improving the known lower bounds requires constructing large independent sets in strong powers of these graphs. Recent AI-assisted work has produced a rapid sequence of improvements: building on the construction of Itty et al., Gao developed a recursive product construction for combining structured independent sets, and Buys, Polak, and Zuiddam (BPZ) subsequently strengthened this through a richer recursion framework. We continue this line of AI-assisted exploration and introduce a heterogeneous refinement of these constructions. The central observation is that the usefulness of an intermediate construction depends not only on the size of its current main independent set, but also on the auxiliary structure it carries into subsequent recursion. Consequently, different parts of that auxiliary structure need not use the same independent set, and different occurrences in a recursion need not use the same intermediate representation. We formalize this for Gao's binary product and derive explicit propagation rules showing how heterogeneous choices strengthen the resulting gadget while leaving its current code size unchanged, then extend the principle to the more general BPZ framework, tailoring constructions to the distinct roles they play within the recursion. Applying these refinements to the seven-cycle $C_7$, we obtain an independent set in $C_7^{\boxtimes 500}$ yielding $Θ(C_7)\ge 3.25883262\ldots$, improving the best known lower bound. Beyond the numerical gain, the results illustrate a general principle for recursive zero-error constructions: intermediate structures with the same dimension and current code size can have different downstream value depending on where and how they are used in the recursion.
