Bh Sets and Costas Cubes

A Costas array is a permutation array in which the vectors joining pairs of 1’s are all distinct. Jedwab and Yen introduced Costas cubes, which are three-dimensional arrays that satisfy the condition that their three two-dimensional projections are Costas arrays. On the other hand, a subset A in an additive group G is a $B_{h}$ set if all sums of the form $a_{1} + a_{2} \cdots + a_{h}$ , where $a_{1}, a_{2} \cdots, a_{h} \in A$ , are different. In this paper, we present three new constructions of $B_{h}$ sets for all $h\geq 3$ . Moreover, in the case $h=3$ , we analyze the relationship of these sets with Costas cubes constructed by Jedwab and Yen, and determine which of these cubes are $B_{3}$ sets.