# 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, by constructing a testing set of size s^n, s<q-1, for frequency n-cubes (a testing sets 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. Keywords: frequency hypercube, correlation-immune function, latin hypercube, testing set.

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