Necessary and Sufficient Conditions for Some two Variable Orthogonal Designs in Order $44$

Abstract

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:
$$\begin{array}{r}(5,34),(8,31),(9,33),(13,29),\\ (7,32),(9,30),(11,30),(15,26)\end{array}$$
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.