A note on cover free families

TL;DR

This paper determines the exact size of cover-free families for specific parameters, advancing understanding in combinatorial design theory and its relation to the Hadamard conjecture.

Contribution

It provides the exact values of N((r,w;d),t) for all r,w with r+w<t+1 and certain d, filling a key gap in combinatorial design knowledge.

Findings

Exact values of N((r,w;d),t) for specified parameters.

Connections established between cover-free families and Hadamard conjecture.

Enhanced understanding of combinatorial structures in design theory.

Abstract

Let N((r,w;d),t) denote the minimum number of points in a (r,w;d)-cover-free family having t blocks. Hajiabolhassan and Moazami (2012)[6] showed that the Hadamard conjecture is equivalent to confirm N((1,1;d)4d-1)=4d-1. Hence, it is a challenging and interesting problem to determine the exact value of N((r,w;d),t). In this paper, we determine the exact value of N((r,w;d),t) for every r, w, where r+w<t+1 and some d.