Quaternary splitting algorithm in group testing

In Classical group testing, one is given a population of n items N which contains some defective d items inside. A group test (pool) is a test on a subset of N. Under the circumstance of no errors, a test is negative if the testing pool contains no defective items and the test is positive if the testing pool contains at least one defective item but we don’t know which one. The goal is to find all defectives by using as less tests as possible, mainly to minimize the number of tests (in the worst case situation). Let M(d, n) denote the minimum number of tests in the worst case situation where |N|=n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|N|=n$$\end{document} and d is the number of defectives. In this paper, we focus on estimating M(d, n) and obtain a better result than known ones in various cases of d and n.