New orthogonal designs and sequences with two and three variables in order 28

We give new sets of sequences with zero autocorrelation function and entries from the set {0, +/-a, +/-b, +/-c} where a, b and c are commuting variables (which may also be set zero). Then we use these sequences to construct some new orthogonal designs. We show the known necessary conditions for the existence of an OD(28; s(1), s(2), s(3)) plus the condition that (s(1), s(2), s(3)) not equal (1, 5, 20) are sufficient conditions for the existence of an OD(28; s(1), s(2), s(3)). We also show the known necessary conditions for the existence of an OD(28; s(1), s(2), s(3)) constructed using four circulant matrices are sufficient conditions for the existence of 4-NPAF(s(1), s(2), s(3)) of length n for all lengths n greater than or equal to 7. We establish asymptotic existence results for OD(4N; s(1), s(2)) for 3 less than or equal to s(1) + S-2 less than or equal to 28. This leaves no cases undecided for 1 less than or equal to s(1) + s(2) less than or equal to 28 We show the known necessary conditions for the existence of an OD(28; s(1), s(2)) with 25 less than or equal to s(1) + s(2) less than or equal to 28, constructed using four circulant matrices, plus the condition that (S-1, s(2)) not equal (1, 26), (2, 25), (7, 19), (8, 19) or (13, 14), are sufficient conditions for the existence of 4-NPAF(s(1), s(2)) of length n for all lengths n greater than or equal to 7.