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 subsetAin an additive groupGis aBhset if all sums of the forma1+a2· · · +ah, wherea1,a2· · · ,ah∈A, are different. In this paper, we present three new constructions ofBhsets for allh≥ 3. Moreover, in the caseh= 3, we analyze the relationship of these sets with Costas cubes constructed by Jedwab and Yen, and determine which of these cubes areB3sets.