An upper bound on the number of frequency hypercubes

TL;DR

The paper addresses tightening upper bounds on the number of frequency $n$-cubes by constructing smaller testing sets. It introduces and analyzes testing sets for $k$-frequency $n$-cubes, linking these objects to bitrades, linear/affine Boolean functions, and error-correcting codes to derive quantitative bounds. For the case $k=1$, it delivers explicit, improved bounds showing the number of cubes grows slower than the trivial bound, with asymptotics $|F_1^n(q;\lambda_0,\dots,\lambda_{m-1})| \le q^{(q-1)^{\alpha_n n}}$ and $\alpha_n \to \frac{1}{3}\log_{q-1}((q-1)^3-1)<1$. The work also extends to $k>1$ via recursive constructions, providing bounds tied to combinatorial objects like $|S(2^{n},k-1)|$ and connections to linear codes, thereby broadening the toolkit for analyzing frequency hypercubes and their generalized forms.

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.

Theorems & Definitions (13)

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