| Let G be a finite abelian group of order v with identity 0.Let λ1,A2,...λm 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(υ,m;k1,...,km;λ1,...,λm)-generalized strong external difference family(briefly(υ,m;k1,...,km;λ1,...,λm)-GSEDF)in G if the multiset equation(?)U △(Di,Dj)=λi(G＼{O})holds for each 1≤ i ≤M.Aυ,m;k1,...,km;λ1,...,λm)-GSEDF is said to be a(υ,m,k,λ)-SEDF when k1 =...=km=k and λ1=...=λm=λ.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(υ,m,k,λ)-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(υ,3;k1,k2,k3;λ1,λ2,λ3)-GSEDF does not exist when k1＋k2 + k3<υ.In addition,we generalize a theorem of Jedwab and Li and give more nonexistence results for(υ,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. |
