On sufficient conditions for some orthogonal designs and sequences with zero autocorrelation function

We give new sets of sequences with entries from {0, ±a, ±b, ±c, ±d} on the commuting variables a, b, c, d and zero autocorrelation function. Then we use these sequences to construct some new orthogonal designs. This means that for order 28 only the existence of the following five cases, none of which is ruled out by known theoretical results, remain in doubt: OD(28; 1, 4, 9, 9), OD(28; 1, 8, 8, 9), OD(28; 2, 8, 9, 9), OD(28; 3, 6, 8, 9), OD(28; 4, 4, 4, 9). We consider 4 - N PAF(Sl, S2, S3, S4) sequences or four sequences of commuting variables from the set {0, ±a, ±b, ±c, ±d} with zero nonperiodic autocorrelation function where ±a occurs Sl times, ±b occurs S2 times, etc. We show the necessary conditions for the existence of an 0D(4n; S1,S2, S3,S4) constructed using four circulant matrices are sufficient conditions for the existence of 4 - NPAF(S1, S2, S3, S4) sequences for all lengths ≥ n, i) for n = 3, with the extra condition (S1,S2,S3,S4) ≠ (1,1,1,9), ii) for n = 5, provided there is an integer matrix P satisfying PPT = diag (S1,S2,S3,S4), iii) for n = 7, with the extra condition that (S1,S2,S3,S4) ≠ (1,1,1,25), and possibly (S1,S2,S3,S4) =I- (1,1,1,16), (1,1,8,18), (1,1,13,13), (1,4,4,9), (1,4,9,9), (1,4,10,10), (1,8,8,9), (1,9,9,9), (2,4,4,18), (2,8,9,9), (3,4,6,8), (3,6,8,9); (4,4,4,9), (4,4,9,9), (4,5,5,9), (5,5,9,9). We show the necessary conditions for the existence of an OD(4n; S1,S2) constructed using four circulant matrices are sufficient conditions for the existence of 4 - NPAF(S1,S2) sequences for all lengths ≥ n, where n = 3 or 5.