關於矩陣的數位方法及錯別性質之研究

Abstract

在這篇論文中，我們探討數位網的數位構造方法，以及可以用來做群試用的鑑別與分離矩陣的推廣。我們首先要探討對於建構數位網和superimposed碼扮演很重要角色的獨立性系統即(d,k,m,s)-系統。如同來線性碼和校對矩陣，數位(t,m,s)-網可以用(d,m,m,s)-系統與組合和數論的論點來構造。 另外，我們介紹鑑別與分離矩陣可容錯的推廣，包括它們之間的關係及迭代構造法。而且我們也可以用(d,k,m,s)-系統來建構superimposed (a,b)-碼,其中a,b是有限制的。除此之外，我們還計算在Johnson結合方案和Grassmann結合方案上的兩個參數e_m,e_{<=m}進而得到(m,1;e_{<=m}/2)-分離及鑑別矩陣。

The digital method for constructing digital nets and the generalizations of disjunct, separable matrices for group testing are studied in this thesis. An independent vector system,called a (d,k,m,s)-system, plays an important role in the constructions of digital nets and superimposed codes. Similar to linear codes and their parity-check matrices, digital (t,m,s)-nets can be constructed in terms of (d,m,m,s)-systems together with some combinatorial and number theoretic arguments. Some generalizations of disjunct and separable matrices for error tolerance will be considered, including some relationships between them, and concatenated constructions for them. We further give constructions of superimposed(a,b)-codes by (d,k,m,s)-systems for some values of a,b. Moreover, the minimum Hamming distances over the Boolean sums of m columns e_m (or at most m columns e_{<=m}) over the Johnson schemes and the Grassmann schemes are evaluated, followed by some (m,1;e_{<=m}/2)-separable matrices and (m,1;e_{<=m}/2)-disjunct matrices as well.