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

We give new sets of sequences with entries from {0, ±a, ±b, ±c} on the commuting variables a, b, c and zero autocorrelation function. Then we use these sequences to construct some new orthogonal de-signs. We show the necessary conditions for the existence of an OD(28; s1, s2, s3) plus the condition that (s1, s2, s3) ≠ (1,5,20) are sufficient conditions for the existence of an OD(28; s1, s2, s3). We also show the necessary conditions for the existence of an OD(28; s1, s2, s3) constructed using four circulant matrices are sufficient conditions for the existence of 4 — NPAF(s1, s2, s3) sequences of of length n for all lengths n ≥ 7. We establish asymptotic existence results for OD(4N; s1, s2) for 2 ≤ s1 + s2 ≤ 28. We show the necessary conditions for the existence of an OD(28; s1, s2) with 25 ≤ s1 + s2 ≤ 28, constructed using four circulant matrices, plus the condition that (s1, s2) ≠ (1,26), (2, 25), (7, 19), (8, 19) or (13, 14), are sufficient conditions for the existence of 4 — NPAF(s1, s2) sequences of of length n for all lengths n ≥ 7.