We give a new algorithm which allows us to construct new sets of sequences with entries from the commuting variables 0, ± a, ± b, ± c, ± d with zero autocorrelation function. We show that for twelve cases if the designs exist they cannot be constracted using four circulant matrices in the Goethals-Seidel array. Further we show that the necessary conditions for the existence of an OD(44; s1, s2) are sufficient except possibly for the following 7 cases: (7, 32) (8, 31), (9, 30) (9, 33) (11,30) (13, 29) (15, 26) which could not be found because of the large size of the search space for a complete search. These cases remain open. In all we find 398 cases, show 67 do not exist and establish 12 cases cannot be constructed using four circulant matrices. We give a new construction for OD(2n) and OD(n + 1) from OD(n). The full OD(44; s1 s2, s3, 44 — s1 — s2 — s3) given in this paper yield at least 68 equivalence classes of Hadamard matrices.