Diameter of a direct power of alternating groups

In this paper we estimate the diameter of a direct power of alternating groups A_k for k ≥ 4. We show that there exists a generating set of minimum size for A_n n > 4 , for which the diameter of A_n is O(n). For k ≥ 5, we show that there exists a generating set of minimum size for A_2^ k , for which the diameter of A_2^ k is at most O(ke(c+1)(log k) 4 log log k ) , for an absolute constant c > 0. Finally for 1 ≤ n ≤ 8, we provide generating sets of size two for A_ 5^k and we show that the diameter of A_5^k with respect to those generating sets is O(n). These results leads us to the sense that the best upper bound known for the diameter of the direct power of non-abelian simple groups (specially alternating groups), i.e. O(n 3 ) [4], may be improved to O(n). Furthermore, these results are more pieces of evidence for a conjecture which has been presented in [8] in 2015.