It is established that the logarithm of the number of latin d-cubes of order n is Θ(n ln n) and the logarithm of the number of pairs of orthogonal latin squares of order n is Θ(n ln n). Similar estimations are obtained for systems of mutually strong orthogonal latin d-cubes. As a consequence, it is constructed a set of Steiner quadruple systems of order n such that the logarithm of its cardinal...

We review results for the embedding of orthogonal partial Latin squares in squares, comparing and contrasting these with squares. also present a new construction that uses existence set $t$ mutually order $n$ to construct $2t$ $n^t$.