Covering Arrays Of Strength ≥3 And Their Related Combinatorial Configurations

Let N, t, k, v,λbe positive integers, where 2≤t≤k. A covering array of size N strength t, degree k, order v and index A, denoted by CAλ(t, k, v), is a N×k array over a v symbol set, such that in every N×t subarray, each t-tuple occurs at leastλtimes. When " at least " is replaced by " exactly ", it defines an orthogonal array denoted by OAλ(t,k,v). The minimum size N for which a CA(t, k, v) exists is called a covering number and written as CAN(t,k,v). The corresponding CA of size N= CAN(t, k, v) is called optimal. Covering arrays including orthogonal arrays as its subclass belong to an important and high-profile area of combinatorics, they have nice applications in statistics, computer science, codes and cryptography, and thus attract considerable attention for long time. For t= 2, much work has been done on CAs and OAs. For t= 3, however, the question becomes very difficult and not too much results are known. In this thesis, we research deeply on CAs of strength equal to or more than 3 and their related combinatorial configurations. The research objects include covering array and covering number, orthogonal arrays, relative difference matrices and ordered orthogonal arrays, which are all of important values both in theory and application.In chapter 2, a bulk of new relative difference matrices including RDMs associated with an adder, RDM*s and CRDMs with five rows are being constructed via cyclotomic theory and Weil theorem on multiplicative character sums cyclotomy.In chapter 3, new constructions of CAs with strength and degree (3,5),(3,6),(4,6) from RDMs associated with an adder and RDM*s were given and the corresponding known upper bounds of covering numbers were improved.In chapter 4, firstly, by using 3BD motheds, the first bulk of OA(3,5,4n+2) in recent years with the smallest n= 62 was given; then the existence of orthogonal array of strength 3, degree 6 and index more than one was observeved and some new OAλ(3,6,v)'s were constructed.In chapter 5, by employing ordered orthogonal arrays, a link between orthogonal of strength 3 with prescribed properties RDOAs and (t, t+3, s)-Nets was given, which generalizes one theorem of Niederreiter and thus give a new constructive method of Nets. Then by using RDOAs, new (t, t+3, s)-Nets were obtained.