An upper bound on the number of frequency hypercubes

A frequency n-cube $F^n(q;l_0,...,l_{m?1})$ is an n-dimensional q-by-...-by-q array, where $q=l_0+...+l_{m?1}$, filled by numbers $0,...,m?1$ with the property that each line contains exactly $l_i$ cells with symbol i, $i=0,...,m?1$ (a line consists of q cells of the array differing in one coordinate). The trivial upper bound on the number of frequency n-cubes is $m^{{(q?1)}^n}$. We improve that lower bound for $n>2$, replacing $q?1$ by a smaller value s, by constructing a testing set of size $s^n$ for frequency n-cubes (a testing set is a collection of cells of an array the values in which uniquely determine the array with given parameters). We also construct new testing sets for generalized frequency n-cubes, which are essentially correlation-immune functions in n q-valued arguments; the cardinalities of new testing sets are smaller than for testing sets known before.