The local bases and bases of primitive anti-symmetric signed digraphs with no loops

Let S be a signed digraph, if the underlying digraph D(S) is symmetric, and each 2-cycle in S is negative, then S is called an anti-symmetric signed digraph. The local bases of primitive anti-symmetric signed digraphs with no loops of order n is studied, and the following conclusion is proved that lS(k)max{n + l-1, n + k-1}, where l is the shortest length of odd cycles of S. The upper bound of the bases of primitive anti-symmetric signed digraphs with no loops of order n are also obtained, and it is shown that the above obtained upper bounds are sharp.