On the existence of mixed orthogonal arrays of strength two with six factors

Orthogonal arrays are foundational objects in combinatorial design theory and experimental design theory, with broad applications in computer science, coding theory, and cryptography. In this paper, we investigate the existence for mixed orthogonal arrays of strength two with six factors, and propose several new constructions based on incomplete orthogonal Latin squares, incomplete transversal designs, and $\mathrm{\Sigma }$-resolvable transversal designs. As a result, we show that the necessary conditions for the existence of such mixed orthogonal arrays are also sufficient, except for a small number of known exceptions and nine unsettled cases.