网络与离散多拟阵的向量线性可解性
On the Vector Linear Solvability of Networks and Discrete Polymatroids
摘要：
我们考虑网络在域$\mathbb{F}_q.$上的向量线性可解性。众所周知，网络在$\mathbb{F}_q$上的标量线性解存在当且仅当网络是关于$\mathbb{F}_q.$可表示的\textit{matroidal}。\textit{discrete polymatroid}是拟阵的多集模拟。本文定义了一个\textit{discrete polymatroidal}网络，证明了网络在域$\mathbb{F}_q$上存在向量线性解的充要条件是网络是关于在$\mathbb{F}_q.$上可表示的离散多拟阵的离散多拟阵。离散多拟阵在$\mathbb{F}_q$上的每一个表示都导致所构造网络在$\mathbb{F}_q$上的向量线性解。给出了构造算法的例子，其中所得到的网络在$\mathbb{F}_q.$上允许向量线性解，但不允许标量线性解
英文摘要：
We consider the vector linear solvability of networks over a field $\mathbb{F}_q.$ It is well known that a scalar linear solution over $\mathbb{F}_q$ exists for a network if and only if the network is \textit{matroidal} with respect to a \textit{matroid} representable over $\mathbb{F}_q.$ A \textit{discrete polymatroid} is the multi-set analogue of a matroid. In this paper, a \textit{discrete polymatroidal} network is defined and it is shown that a vector linear solution over a field $\mathbb{F}_q$ exists for a network if and only if the network is discrete polymatroidal with respect to a discrete polymatroid representable over $\mathbb{F}_q.$ An algorithm to construct networks starting from a discrete polymatroid is provided. Every representation over $\mathbb{F}_q$ for the discrete polymatroid, results in a vector linear solution over $\mathbb{F}_q$ for the constructed network. Examples which illustrate the construction algorithm are provided, in which the resulting networks admit vector linear solution but no scalar linear solution over $\mathbb{F}_q.$
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