Some Results On Generalized Strong External Difference Families

Let G be a finite abelian group of order v with identity 0.Let ?1,A2,...?m be positive integers and let D1,D2,...,Dm be mutually disjoint subsets of G with|D| =ki,1 ? i ?m.Define ?(Di,Dj)= {x-y:x ?Di,y € Dj},1? i,j ?m.Then {D1,D2,...Dm} is called a(?,m;k1,...,km;?1,...,?m)-generalized strong external difference family(briefly(?,m;k1,...,km;?1,...,?m)-GSEDF)in G if the multiset equation(?)U ?(Di,Dj)=?i(G?{O})holds for each 1? i ?M.A?,m;k1,...,km;?1,...,?m)-GSEDF is said to be a(?,m,k,?)-SEDF when k1 =...=km=k and ?1=...=?m=?.GSEDFs were first introduced by Paterson and Stinson in 2016.They can be used to obtain R-optimal strong algebraic manipulation detection codes.Reseachers mainly pay attention to their existence and nonexistence results.So far,many nonexistence results,of(?,m,k,?)-SEDFs have been obtained.Martin and Stinson,and Jedwab and Li separately proved that there are no nontrivial SEDFs with m = 3,4 by using different methods.However,there are only a few of existence results with m = 2 via cyclotomic classes.For m>5 there is only a nontrivial example(243,11,22,20)-SEDF.In this thesis,we use number theory and character theory to prove nonexistence of some classes of GSEDFs.Especially,we prove that a(?,3;k1,k2,k3;?1,?2,?3)-GSEDF does not exist when k1?k2 + k3<?.In addition,we generalize a theorem of Jedwab and Li and give more nonexistence results for(?,2,f,A)-SEDFs.We also give some direct constructions for GSEDFs and use combinatorial methods to give a new recursive construction for GSEDFs.Thus we get some infinite classes of GSEDFs with m = 2,3.