An upper bound on the number of frequency hypercubes

TL;DR

This paper improves the upper bounds on the number of frequency hypercubes by constructing smaller testing sets, which better determine these arrays and their generalized forms, advancing understanding in combinatorial array theory.

Contribution

It introduces new smaller testing sets for frequency hypercubes and their generalizations, improving bounds on their enumeration.

Findings

Constructed testing sets smaller than previous bounds

Improved upper bounds for frequency hypercube counts for n>2

Extended results to generalized frequency hypercubes and correlation-immune functions

Abstract

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.