发表机构
Princeton University（普林斯顿大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文通过构造一个在PG(2,9)上的182顶点二部子图中的确定性线性网络编码，反驳了无向多重单播猜想，证明了网络编码在多播吞吐量上优于多商品流，并引入了基于局部可验证证书的非确定性网络编码模型。
AI 中文摘要
无向多重单播猜想[LL04]断言网络编码相对于多商品流在吞吐量上没有优势。我们通过构造一个在PG(2,9)的点线关联图的182顶点二部子图上的确定性线性网络编码（定义在F_9上）来反驳这一猜想。该构造支持157个独立单播会话，公共编码速率至少为1，而每个分数多商品流的公共速率至多为147/157。根据[BGS17]的放大定理，这产生一族无向多重单播实例，其编码间隙为Ω((log n)^ε)，其中ε>0。我们还引入了一种基于局部可验证证书的网络编码非确定性模型，该模型指导了我们的构造，并且可能具有独立的研究兴趣。基于[BH25]的高围长图和纠错码框架，我们使用GPT-6寻找一个基于新的Reed-Solomon局部码选择的非确定性反例，然后通过边定向和局部搜索将该示例转换为因果码。
英文摘要
The undirected multiple-unicast conjecture [LL04] asserts that network coding offers no throughput advantage over multicommodity flow. We refute this conjecture by constructing a deterministic linear network code over $\mathbb{F}_9$ on a 182-vertex bipartite subgraph of the point-line incidence graph of $\mathrm{PG}(2,9)$. The construction supports $157$ independent unicast sessions at common coding rate at least $1$, while every fractional multicommodity flow has common rate at most $147/157$. By the amplification theorem of~[BGS17], this yields a family of undirected multiple-unicast instances with coding gap $Ω((\log n)^\varepsilon)$ for some $\varepsilon>0$. We also introduce a nondeterministic model of network coding based on locally verifiable certificates, which guides our construction and may be of independent interest. Building on the high-girth graph and error-correcting code framework of [BH25], we use GPT-6 to find a nondeterministic counterexample based on a new choice of Reed--Solomon local codes, and then convert this example into a causal code using an edge orientation and local search.
