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.

Cite: Krotov D.S. , Potapov V.N.
An upper bound on the number of frequency hypercubes
Discrete Mathematics. 2024. V.347. N1. 113657 :1-9. DOI: 10.1016/j.disc.2023.113657 WOS Scopus РИНЦ OpenAlex

Dates:

Submitted:	Jan 18, 2023
Accepted:	Aug 4, 2023
Published online:	Aug 25, 2023
Published print:	Apr 2, 2024

Identifiers:

Web of science:	WOS:001068924600001
Scopus:	2-s2.0-85168847295
Elibrary:	64863281
OpenAlex:	W4385645329

Citing:

DB	Citing
Scopus	1

Altmetrics: