RGDDs, Perfect Difference Families And Conflict-Avoiding Code

In this article, we mainly discuss some important questions in combinatorial designs and coding theories, it includes resolvable group divisible designs(RGDDs), conflict-avoiding code(CACs) and perfect difference family(PDFs).The paper is organized as follows.Chapter 1 investigates the existence of RGDD with block size four, group-type hn and indexλ= 2, we show that these necessary conditions are sufficient for allλ> 2, with the definite exceptions of (λ, h, n)∈{(3,2,6)}U{(2j+1,2,4): j>1}. The known existence result forλ= 1 is also improved.In Chapter 2, we discuss perfect difference families(PDF), which can be used to apply in Radar Array and CDMA system. Research on perfect difference fam-ilies began from 1970's, the existence result we known about PDF(12t+1,4,1) is t= 1,4 - 33,36,41 when t< 50. In this paper, by some new method, we improve the result to that there exist PDF(12t+1,4,1) for t= 1,4-1000.In the last Chapter of this article, we consider the construction on optimal conflict-avoiding code(CAC), which is used in multiple-access channel without feedback. It has been known that lots of constructions of optimal CAC(n, k) exist when k= 3,4 and 5, We will give the upper bound on size k= 6 and construct a new infinite series CACs which attend the upper bound.