Matroid theory is fundamentally connected with index coding and network coding problems. In fact, the reliance of linear index coding and network coding rates on the characteristic of a field has been demonstrated by using the two well-known matroid instances, namely the Fano and non-Fano matroids. This established the insufficiency of linear coding, one of the fundamental theorems in both index coding and network coding. While the Fano matroid is linearly representable only over fields with characteristic two, the non-Fano instance is linearly representable only over fields with odd characteristic. For fields with arbitrary characteristic $p$, the Fano and non-Fano matroids were extended to new classes of matroid instances whose linear representations are dependent on fields with characteristic $p$. However, these matroids have not been well appreciated nor cited in the fields of network coding and index coding. In this paper, we first reintroduce these matroids in a more structured way. Then, we provide a completely independent alternative proof with the main advantage of using only matrix manipulation rather than complex concepts in number theory and matroid theory. In this new proof, it is shown that while the class $p$-Fano matroid instances are linearly representable only over fields with characteristic $p$, the class $p$-non-Fano instances are representable over fields with any characteristic other than characteristic $p$. Finally, following the properties of the class $p$-Fano and $p$-non-Fano matroid instances, we characterize two new classes of index coding instances, respectively, referred to as the class $p$-Fano and $p$-non-Fano index coding, each with a size of $p^2 + 4p + 3$.

翻译：拟阵理论与索引编码及网络编码问题存在根本性联系。事实上，线性索引编码与网络编码速率对域特征的依赖性已通过两个著名的拟阵实例——即Fano拟阵与非Fano拟阵——得以证明。这确立了线性编码的不足性，成为索引编码与网络编码领域的基本定理之一。Fano拟阵仅能在特征为二的域上线性表示，而非Fano实例仅能在奇特征域上线性表示。针对任意特征$p$的域，Fano与非Fano拟阵被推广至新的拟阵实例类，其线性表示依赖于特征为$p$的域。然而，这些拟阵在网络编码与索引编码领域尚未得到充分重视或引用。本文首先以更结构化的方式重新引入这些拟阵，随后提供一个完全独立的替代证明，其主要优势在于仅使用矩阵操作而非数论与拟阵理论中的复杂概念。该新证明表明：$p$-Fano拟阵实例类仅能在特征为$p$的域上线性表示，而$p$-非Fano实例类可在除特征$p$外的任意特征域上表示。最后，基于$p$-Fano与$p$-非Fano拟阵实例类的性质，我们刻画了两类新的索引编码实例，分别称为$p$-Fano索引编码类与$p$-非Fano索引编码类，其规模均为$p^2 + 4p + 3$。