Constructions of pairs of orthogonal latin cubes

A pair of orthogonal latin cubes of order $q$ is equivalent to a maximum distance separable code with distance 3 or to an ${\mathit{OA}}_1\left(3,5,q\right)$ orthogonal array. We construct pairs of orthogonal latin cubes for sequences of previously unknown orders $q_i=16\left(18i?1\right)+4$ and $q_i^{\prime }=16\left(18i+5\right)+4$. The minimal new obtained parameters of orthogonal arrays are ${\mathit{OA}}_1\left(3,5,84\right)$.