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In \cite{Sz17-2} we considered hyperball packings in $3$-dimensionalhyperbolic space. We developed a decomposition algorithm that for eachsaturated hyperball packing provides a decomposition of $\HYP$ into truncatedtetrahedra. In order to get a density upper bound for hyperball packings, it issufficient to determine the density upper bound of hyperball packings intruncated simplices. Therefore, in this paper we examine the doubly truncatedCoxeter orthoscheme tilings and the corresponding congruent and non-congruenthyperball packings. We proved that related to the mentioned Coxeter tilings thedensity of the densest congruent hyperball packing is $\approx 0.81335$ that is-- by our conjecture -- the upper bound density of the relating non-congruenthyperball packings too.

Congruent and non-congruent hyperball packings related to doubly truncated Coxeter orthoschemes in hyperbolic $3$-space