We investigate the structure of intersecting error-correcting codes, with a particular focus on their connection to matroid theory. We establish properties and bounds for intersecting codes with the Hamming metric and illustrate how these distinguish the subfamily of minimal codes within the family of intersecting codes. We prove that the property of a code being intersecting is characterized by the matroid-theoretic notion of vertical connectivity, showing that intersecting codes are precisely those achieving the highest possible value of this parameter. We then introduce the concept of vertical connectivity for $q$-matroids and link it to the theory of intersecting codes endowed with the rank metric.

翻译：本文研究了相交纠错码的结构，特别关注其与拟阵理论的联系。我们建立了汉明度量下相交码的性质与界，并阐明了这些性质如何将极小码子族从相交码族中区分出来。我们证明了码的相交性可由拟阵理论中的垂直连通性概念来刻画，表明相交码恰好是达到该参数最大可能值的码。随后，我们引入了$q$-拟阵的垂直连通性概念，并将其与赋予秩度量的相交码理论联系起来。