The weighing matrices of order 4n and weight 4n-2 and 2n-1

We give algorithms and constructions for mathematical and computer searches which allow us to establish the existence of W(4n, 4n - 2) and W (4n, 2n - 1) for many orders 4n less than 4000. We compare these results with the orders for which W(4n, 4n) and W(4n, 2n) are known. We use new algorithms based on the theory of cyclotomy to obtain new T-matrices of order 43 and JM-matrices which yield W(4n, 4n - 2) for n = 5,7,9,11,13,17,19,25,31,37,41,43,61,71,73,157.