External Difference Families,Difference Set Pairs And Generalized Relative Difference Set Pairs

In 2004, Wakaha Ogata et al. introduced a new combinatorial design calledexternal difference families (EDF for short), and showed that by using EDF one canconstruct authentication codes with arbitration,optimal splitting A-codes andsecret sharing schemes and so on. In 2006, Chang Y and Ding C discussed EDF bymeans of group, cyclotomic class and recursive constructions.Xu cheng qian introduced a new combinatorial design called difference set pairs(DSP for short), and showed that by using DSP one can construct the perfect binaryarray pairs in paper[9]. Then Xu cheng qian et al. generalized DSP to generalizedrelative difference set pairs.This thesis consists of three chapters. In chapter one, we present the nessarydefinitions and give the survey of known results of external difference families,difference set pairs and generalized relative difference set pairs. In chapter two,the nonexistence of a class of EDF is proved by using Hall polynomial, this showsthat over Z_v, the necessary condition for the existence of EDF proposed by WakahaOgata et al. is insufficient. Then we discuss the possibility of constructing EDF byusing cyclotomy classes of order 4 and 6, and obtain some new classes of externaldifference families. In chapter three, we construct some new classes of differenceset pairs and generalized relative difference set pairs by using cyclotomy modulo pq,where p and q are distinct primes, and construct some new classes of generalizedrelative difference set pairs by direct product.