On the number of frequency hypercubes $F^n(4;2,2)$

Abstract

A frequency n-cube F n (4; 2, 2) is an n-dimensional 4 × • • • × 4 array filled by 0s and 1s such that each line contains exactly two 1s. We classify the frequency 4-cubes F 4 (4; 2, 2), find a testing set of size 25 for F 3 (4; 2, 2), and derive an upper bound on the number of F n (4; 2, 2). Additionally, for any n greater than 2, we construct an F n (4; 2, 2) that cannot be refined to a latin hypercube, while each of its sub-F n−1 (4; 2, 2) can.