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A family $\mathcal{F}$ of subsets of a set $X$ is $t$-intersecting if $\vertA_i \cap A_j \vert \geq t$ for every $A_i, \; A_j \in \mathcal{F}$. We studyintersecting families in the Hamming geometry. Given $X=\mathbb{F}_q^3$ avector 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 inthe scalar multiples of a vector. The geometric behavior of extended balls isdiscussed. As the main result, we investigate a ``large" arrangement of vectorswhose extended balls are ``highly intersecting". Consider the followingcovering problem: a subset $\mathcal{H}$ of $\mathbb{F}_q^3$ is a shortcovering 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, minimalcardinality of a short covering is improved for some instances of $q$.

Intersecting families of extended balls in the Hamming spaces