The Constructions Of(q,K,λ,t,Q) Almost Difference Families

The definition of(q,k,λ,t)-almost difference set(ADS for short) was introduced by Davis in1992,in which g,k,λ,t are all positive integers.The notion of a(q,k,λ,t)-almost difference family(ADF for short)was introduced and studied by Ding and Yin as an useful generalization of the notion of a(g,k,λ,t)-ADS,and a few constructions on(q.k,λ,t)-ADS via finite fields were also presented. The characteristic sequences of a cyclic ADF and its shifts form a collectlon of binary sequences with optimal autocorrelation.and hence they have many applications in communication and stream ciphers, So it is worth to studied (q,K,λ,t)-ADFs.In this thesis we consider,more generally,(q,K,λ,t)-ADFs,where K={k1,k2,...,kr｝is a set of positive integers and Q=(q1,q2,...,qr)is a given block-size distribution sequence.Let F={B1,B2,...,Bs｝be a family of subsets of an Abelian group G of order q and K={|Bi|:1≤i≤s］).Define the difference list△Bj of Bj(1≤j≤s)to be the multi-set,△Bj={a一b:a,b∈Bj and a≠b］,1≤j≤s,and the difference list△F of to be the multi-set△F=U1≤j≤s△Bj,F is sald to be a(q,K,λ,t,Q)almost difference family ((q,K,λ,t.Q)一ADF for shrot)if some t nonzero elements of G occur exactly λ times cach in the difference list△F,while the remaining q-1-t nonzero elements of G occur exactly λ+1times each in△F,and qi is the ratio between the total number of blocks of F and the number of those of size ki.If G is cyclic,then we call the(q,K,λ,t.Q)-ADF cyclic.We say that Q is normalized if it is written in the form with gcd(a1,a2,...,ar,)=1. It is clear that∑ai=b for normalized Q. By a balanced (q.K,λ,t,Q)-ADF we mean a(q,K,λ,t,Q)-ADF with Q=(1/r,1/r,...,1/r),namely an ADF in which the number of blocks of a given size is a constant. The following is a necessary condition for the existence of a(q,K,λ,t,Q)一ADF.Theorem1.1The necessary condition for the existence of a(q,K,λ.t,Q)-ADF with normalized distribution sequence Q=(a1/b,a2/b,...,ar/b)is that (λ+1)(q-1)三t(mod ω), where It is clear that for any subset B of Fq,there exists a mutiset B such that△B=BU(一1)B.If B is a collection of subsets of Fq,then we set B:UB∈8B.In this thesis,we have the following construction for(q,K,λ,t,Q)-ADF.Theorem1.2Let q≡1(mod2e)be a prime,K={k1,k2,...,kr)a set of positive integers,Q=(a1/b,a2/b,...,ar/b)be normalized,and B be a collection of b subsets of Fq such that there exactly ni blocks of size ki in B,for Assume that B has exa.ctly λ+1elements(counting multiplicities)in each of s cyclotomic classes of index e and exactly λ elements in each of the remaining e—s classes.Then there exists a cyclic(q,K,λ,t,Q)一ADF witht=(q-1)(e-s)/e.In this thesis,by using theorem1.2and computer searching,the following results arc obtained.Theorem1.3for each odd prime q≡1(mod4),there exists a cyclic balanced (q,{3,4｝,4,(q-1)/2)-ADF in FqTheorem1.4for each odd prime q≡1(mod8),there exists a cyclic balanced (q,{3,4｝,2,3(q-1)/4)-ADF in Fq.Theorem1.5for each odd prime q≡1(mod10),there exists a cyclic balanced (q,{3,4｝,1,(g-1)/5)-ADF in FQ.Theorem1.6for each odd prime q≡1(mod12),there exists a.cyclic balanced (g:{3:4｝,1,(q-1)/2)-ADF in Fq.Theorem1.7for.each odd primee q≡1(mod4),there exists a cyclic balanced (q.{3,5｝,6,(q-1)/2)-ADF in Fq.Theorem1.8for each odd prime q≡1(mod6),there exists a cyclie balanced (q,{3,5｝,4,2(q-1)/3)-ADF in Fq.Theorem1.9for each odd prime q≡1(mod8),there exists a cyclic balanced (q:{3,5｝,3,3(q-1)/4)-ADF in Fq.Theorem1.10for each odd prime q≡1(mod10),there exists a cyclic balanced (q,{3,5｝,2,2(g-1)/5)-ADF in Fq.Theorem1.11for each odd prime q≡1(mod4),there exists a cyclic balanced (q,{3,4,5｝,9,(q-1)/2)-ADF in Fq.Theorem1.12for each odd prime q≡1(mod6):there exists a cyclic ba1anced (q:{3,4,5｝,6,2(q-1)/3)-ADF in Fq.Theorem1.13for each odd prime q≡1(mod10), there exists a cyclic(q,{3,4｝,2.3(q-1)/5,(2/3,1/3))-ADF in Fq. Theorem1.14for each odd prime q≡1(mod14),there exists a cyclic(q,{3,4},1,2(q-1)/7,(2/3,1/3))-ADF in Fq.This thesis is divided into six parts.In chapter one,we present some notations,the nec-essary condition for the existence of a(q,K,λ,t,Q)-ADF and a construction of (q,k,λ,Q)-ADFs. Chapter§2-5focus the constructions of balanced cyclic(q,K,λ)-ADFs with ω∈{{3,4},{3,5},{3,4,5}},and cyclic(q,{3,4},λ,(2/3,1/3))-ADFs,Concluding remarks and problems for further research are given in Chapter six.