In this paper we construct a new family of lattice packings for superballs in three dimensions (unit balls for the $l^p_3$ norm) with $p in (1, 1.58]$. We conjecture that the family also exists for $p in (1.58, log_2 3 = 1.5849625ldots]$. Like in the densest lattice packing of regular octahedra, each superball in our family of lattice packings has $14$ neighbors.
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