In 1992, Brualdi and Jung first introduced the maximum jump number M(n, k), that is, the maximum number of the jumps of all (0, 1)-matrices of order n with k 1's in each row and column, and then gave a table about the values of M(n, k) when 1 ≤ k ≤ n ≤ 10. They also put forward several conjectures, including the conjecture M(2k - 2, k) = 3k - 4 + [(k-2)/2]. In this paper, we prove that b(A) ≥ 4 for every A ∈Λ(2k - 2, k) if k ≥ 11, and find another counter-example to this conjecture . |