on the spectrum of $r$-orthogonal latin squares of different orders

two latin squares of order $n$ are orthogonal if in their superposition, each of the$n^{2}$ ordered pairs of symbols occurs exactly once. colbourn, zhang and zhu, in a seriesof papers, determined the integers $r$ for which there exist a pair of latin squares oforder $n$ having exactly $r$ different ordered pairs in their superposition. dukes andhowell defined the same problem for latin squares of different orders $n$ and $n + k$.they obtained a non trivial lower bound for $r$ and solved the problem for $k geq‎ 2n/3$.here for $k < 2n/3$, some constructions are shown to realize many values of $r$ and forsmall cases ($3 leg n leq 6$), the problem has been solved.