  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 $p \in (1.58, \log_2 3 =
1.5849625\ldots]$. Like in the densest lattice packing of regular octahedra,
each superball in our family of lattice packings has $14$ neighbors.
