Intersecting families of extended balls in the Hamming spaces

TL;DR

This paper investigates the structure and intersection properties of extended balls in Hamming spaces, providing new bounds on minimal short coverings and analyzing their geometric behavior in finite vector spaces.

Contribution

It introduces a detailed study of intersecting families of extended balls in finite Hamming spaces and improves bounds on minimal short coverings for certain parameters.

Findings

Characterization of intersecting families of extended balls

New bounds on minimal short coverings in finite fields

Analysis of the geometric behavior of extended balls

Abstract

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 $q$.