AI 中文总结
本文针对广义组合网络和Zosin-Khuller网络，建立标量与向量网络MDS码的系统框架，推导最小域大小的更优界，明确一类向量编码无域大小优势的网络，并界定标量与向量解的MDS差距。
AI 中文摘要
本文研究网络最大距离可分（MDS）码所需的最小域大小，该参数是影响网络节点计算复杂度的关键因素。针对广义组合网络和Zosin-Khuller网络，我们为标量和向量网络MDS码构建了一套系统框架。对于广义组合网络上的标量码，我们建立了网络码的最小距离与经典线性码的最小汉明距离之间的等价关系，将网络级MDS约束转化为编码理论条件，得到了与经典MDS码和覆盖格拉斯曼码相关的充要存在条件。通过改进的贪心构造和MRD码设计，我们得到了最小域大小的更优界，优于现有的通用界。对于向量网络码，我们建立了类似的距离等价关系，并通过覆盖格拉斯曼码刻画其存在性，得到了最小有效域大小的界。值得注意的是，在若干参数范围内，最优标量和向量MDS码之间的差距消失；我们明确确定了一类网络，其中向量编码在域大小上不优于标量编码。对于Zosin-Khuller网络，我们利用超图同态和子集相交论证推导了向量MDS码的最小有效域大小的下界，严格改进了现有的标量界。我们还基于超图同态提供了向量MDS构造的充要条件，得到了最小域大小的上界。最后，我们对最优标量和向量解之间的MDS差距进行了界定。
英文摘要
This paper investigates the minimum field size required for network maximum distance separable (MDS) codes, a critical parameter affecting computational complexity at network nodes. Focusing on generalized combination networks and Zosin Khuller networks, we develop a systematic framework for both scalar and vector network MDS codes. For scalar codes on generalized combination networks, we establish an equivalence between the minimum distance of network codes and the minimum Hamming distance of classical linear codes, converting network-level MDS constraints into coding theory conditions. This yields necessary and sufficient existence conditions linked to classical MDS codes and covering Grassmannian codes. Using refined greedy constructions and MRD code designs, we obtain improved bounds on the minimum field size, outperforming the prior universal bound. For vector network codes, we develop an analogous distance equivalence and characterize existence via covering Grassmannian codes, yielding bounds on the minimum effective field size. Notably, the gap between optimal scalar and vector MDS codes vanishes for several parameter regimes; we explicitly identify a family of such networks where vector coding offers no field size advantage over scalar coding. For Zosin Khuller networks, we derive lower bounds on the minimal effective field size of vector MDS codes using hypergraph homomorphisms and subset intersection arguments, strictly improving prior scalar bounds. We further provide a necessary and sufficient condition built upon hypergraph homomorphisms for vector MDS construction, yielding an upper bound on the minimum field size. Finally, we bound the MDS gap between optimal scalar and vector solutions.
