Intersecting families of extended balls in the Hamming spaces

• Published in: arXiv.org
• Main Authors: Martinhão, Anderson N, Emerson L Monte Carmelo
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 01.04.2014
• Subjects: Fields (mathematics)
• ISSN: 2331-8422

A family \(\mathcal{F}\) of subsets of a set \(X\) is \(t\)-intersecting if \(\vert A_i \cap A_j \vert \geq t\) for every \(A_i, \; A_j \in \mathcal{F}\). We study intersecting families in the Hamming geometry. Given \(X=\mathbb{F}_q^3\) a vector space over the finite field \(\mathbb{F}_q\), consider a family where each \(A_i\) is an extended ball, that is, \(A_i\) is the union of all balls centered in the scalar multiples of a vector. The geometric behavior of extended balls is discussed. As the main result, we investigate a ``large" arrangement of vectors whose extended balls are ``highly intersecting". Consider the following covering problem: a subset \(\mathcal{H}\) of \(\mathbb{F}_q^3\) is a short covering if the union of the all extended balls centered in the elements of \(\mathcal{H}\) is the whole space. As an application of this work, minimal cardinality of a short covering is improved for some instances of \(q\).