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 $4$-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

v.2: revised; computational data attached

References in corpus (4)

• The On-Line Encyclopedia of Integer Sequences
• n-Ary quasigroups of order 4
• Asymptotics for the number of n-quasigroups of order 4
• A Lower Bound on the Number of Boolean Functions with Median Correlation Immunity

Cited by in corpus (2)

• Characterization of polystochastic matrices of order $4$ with zero permanent
• An upper bound on the number of frequency hypercubes