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We give a new algorithm which allows us to construct new sets of sequences with entries from the commuting variables \(0, \pm a, \pm b\), with zero autocorrelation function.
We show that for eight cases if the designs exist they cannot be constructed using four circulant matrices in the Goethals-Seidel array. Furthermore, we show that the necessary conditions for the existence of an \(\text{OD}(44; s_1, s_2)\) are sufficient
except possibly for the following \(8\) cases:
\begin{align*}
(5,34), (8,31), (9,33), (13,29),\\
(7,32), (9,30), (11,30), (15,26)
\end{align*}
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 \(399\) cases, show \(67\) do not exist
and establish \(8\) cases cannot be constructed using four circulant matrices.
We give a new construction for \(\text{OD}(2n)\) and \(\text{OD}(n+1)\) from \(\text{OD}(n)\).
We note that all \(\text{OD}(44; s_1, 44-s_2)\) are known except for \(\text{OD}(44; 16, 28)\). These give \(21\) equivalence classes of Hadamard matrices.