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-qarray, 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. (c) 2023 Elsevier B.V. All rights reserved.