On the Smith normal form of skew E-W matrices

Let t be a positive integer. An E-W matrix is a square (-1, 1)-matrix of order 4t + 2 satisfying that the absolute value of its determinant attains Ehlich-Wojtas' bound. We show that the Smith normal form of every skew E-W matrix follows this pattern diag[1, 2, ... , 2, m(2t+2),m(2t+3), ... , m(4t+2)], where m(2t+3) > 2 and the product m(1) ... m(k) divides 2(t)(2[k/2])([k/2]), for 1 <= k <= 4t.