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 \(\{0, \pm a, \pm b, \pm c\}\) 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 \((s_1, s_2, s_3) \neq (1, 5, 20)\) 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 2 for all lengths \(n \geq 7\).

We establish asymptotic existence results for OD\((4N; s_1, s_2)\) for \(3 \leq s_1 + s_2 \leq 28\). This leaves no cases undecided for \(1 \leq s_1 + s_2 \leq 28\). We show the known necessary conditions for the existence of an OD\((28; s_1, s_2)\) with \(25 \leq s_1 + s_2 \leq 28\), constructed using four circulant matrices, plus the condition that \((s_1, s_2) \neq (1, 26), (2, 25), (7, 19), (8, 19)\) or \((13, 14)\), are sufficient conditions for the existence of 4-NPAF\((s_1, s_2)\) of length \(n\) for all lengths \(n \geq 7\).