A new construction for Hadamard matrices of orders 8t + 4

AbstractIn Hadamard matrices of orders 8t + 4, there are usually four rows which agree on exactly one column. In fact, for t = 0, 1, 2 such a “Hall set” always occurs. This is obvious for t = 0, 1 and Hall has shown this for t = 2. When t = 3, the evidence indicates that nearly all H(28) have a Hall set. (Nearly the opposite seems to be true for matrices H(8t).) If a Hall set is assumed to exist for some H and some t, the remaining rows fall into 4 sets which determine 16 submatrices of order t. Several well-known techniques may be applied to such a configuration, and give immediate examples for t = 1, 2, 3, 4