Intersecting families of extended balls in the Hamming spaces

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A family F of subsets of a set X is t-intersecting if Aᵢ ∩ Aⱼ ≥ t for every Aᵢ, Aⱼ ∈ F. We study intersecting families in the Hamming geometry. Given X=Fq³ a vector space over the finite field Fq, consider a family where each Aᵢ is an extended ball, that is, Aᵢ 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 H of Fq³ is a short covering if the union of the all extended balls centered in the elements of H is the whole space. As an application of this work, minimal cardinality of a short covering is improved for some instances of q.