An orthogonal array of index unity, or order v, degree k and strength t, denoted by OA(t, k, v), is a \(k\times v^t\) array with entries from a set V of \(v\ge 2\) symbols such that each of its \(t\times v^t\) subarrays contains every \(t\times 1\) column vector over V exactly once. The construction of orthogonal arrays OA(3, 6, v) remains a longstanding open problem, particularly for values of \(v \equiv 2\pmod {4}\) or \(v \equiv 3,15 \pmod {18}\) . In this paper, we extend the framework of three dimensional orthogonal large sets of disjoint incomplete Latin squares, and develop new algebraic constructions over finite fields. As a result, we establish new infinite families of OA(3, 6, v), significantly expanding the known parameter spectrum for these arrays.
