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

TL;DR

This paper studies the structure and classification of frequency hypercubes with specific line constraints, providing new classifications, bounds, and constructions that distinguish them from Latin hypercubes.

Contribution

It classifies frequency 4-hypercubes, finds a minimal testing set for frequency 3-hypercubes, and constructs hypercubes that cannot be refined into Latin hypercubes.

Findings

Classified frequency 4-hypercubes $F^4(4;2,2)$

Identified a testing set of size 25 for $F^3(4;2,2)$

Derived an upper bound on the number of $F^n(4;2,2)$

Abstract

A frequency $n$-cube $F^n(4;2,2)$ is an $n$-dimensional $4$-by-...-by-$4$ array filled by $0$s and $1$s such that each line contains exactly two $1$s. 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. Keywords: frequency hypercube, frequency square, latin hypercube, testing set, MDS code