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.