On generalizations of separating and splitting families

TL;DR

This paper introduces and analyzes generalizations of separating and splitting families, providing bounds on their minimal sizes and exploring their properties and connections to existing concepts.

Contribution

It defines $n$-separating and $n$-splitting families, studies their properties, and derives bounds on their minimal sizes, extending the theory of these combinatorial structures.

Findings

Bounds on minimal sizes of $n$-separating families are asymptotically tight.

Partial results and open questions are provided for $n$-splitting families.

Connections to classical separating and splitting families are established.

Abstract

The work in this article is concerned with two different types of families of finite sets: separating families and splitting families (they are also called "systems"). These families have applications in combinatorial search, coding theory, cryptography, and related fields. We define and study generalizations of these two notions, which we have named $n$-separating families and $n$-splitting families. For each of these new notions, we outline their basic properties and connections with the well-studied notions. We then spend the greatest effort obtaining lower and upper bounds on the minimal size of the families. For $n$-separating families we obtain bounds which are asymptotically tight within a linear factor. For $n$-splitting families this appears to be much harder; we provide partial results and open questions.