Intersecting families of extended balls in the Hamming spaces

Abstract

A family $\mathcal{F}$ of subsets of a set $X$ is $t$-intersecting if $|A_i\cap A_j|\ge t$ 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 $q$.

19 pages