t x (t+1) d-可分離矩陣的最小t值: d=2或3的情況

Abstract

群試設計(group testing)為應用數學的一個分支，其應用層面包含了錯誤更正碼、基因(DNA)測試等。本論文著重在探討t x (t+1) d-可分離群式設計的可能性。首先我們考慮投影平面的點線關係矩陣，並證明刪除任一列可以產生t x (t+1) d-可分離群矩陣，當t等於d^2+d且d為質數的次方。接著我們證明當t小於d^2+d且d為2或3時並不存在d-可分離群矩陣。

Group testing is a branch of applied mathematics and has several applications, such as error correcting codes, DNA testing, etc. This thesis investigates the existence of a t x (t+1) d-separable matrix for some t and d.

First, we consider the point-block incidence matrix of the projective plane of order d and show that removing any row from the matrix yields a t x (t+1) d-separable matrix as t=d^2+d and d is a prime power. Then, we show that if t<d^2+d and d=2 or 3, there is no t x (t+1) d-separable matrix.