Cyclotomic Constructions Of External Difference Families And Disjoint Difference Families

Difference families are one of most commonly method of constructing balanced incompleteblock design (BIBD).It had been well studied and had applications in coding theory and cryptog-raphy. In 2004, Ogata, Kurosawa, Stinsion and Saido [1] introduced a new type of combinatorialdesigns——external difference families(EDF), which are related to difference families and used itto construct authentication codes and secret sharing.In 2006, Chang Yanxun and Ding Cunsheng [2] presented some results of external differencefamilies and disjoint difference families by some earlier ideas of recursive and cyclotomic construc-tions. The following research problems are presented in[2].Problem 1 Construct other (v, (v-1)/2,(v-1)/2; 2)-EDFs.Problem 2 Give more constructions of (v, k, k-1)-DDFs in Abelian groups G.Problem 4 Find more constructions of (v, k,λ; u)-EDFs in Abelian groups G with ku＜v-1.In this paper, some results of external difference families and disjoint difference families areobtained by cyclotomic construction. At the same time, Theorems [2.10, 3.5], Theorems [2.11,3.5, 3.7], and Theorems [2.12, 3.2, 3.3, 3.6] giving an answer to research problem 1, 2 and 4 in[2], respectively.Let q be a power of an odd prime, andαbe a generator of GF(q)^*=GF(q)\{0}. Assumethat q-1=el, where e＞1 and l＞1 are integers. Define C₀^(e)={α^es|s=0, 1,..., f-1} to bethe subgroup of GF(q)^* generated byα^e, and C_i^(e)=αⁱC₀^(e)={α^es+i|s=0,1,...,f-1} for eachi with 0≤i≤e-1. These C_i^(e) are called cyclotomic classes of order e with respect to GF(q)^*.The cyclotomic numbers of order e, denoted (i,j)_e, are defined as the numbers of the solution ofthe equation z_i+1=z_j, z_i∈C_j^(e),0≤i,j≤e-1.the following results are obtained.Theorem 2.3 Let q=ef+1 is an odd prime power, D_j=C_j1^(e)∪C_j2^(e)∪...∪C_jl^(e),1≤j≤h,then D={D₁, D₂, ..., D_h} is a disjoint difference families over GF(q) if and only if J₀=J₁=…=J_e-1where J_s=sum from j=1 to h sum from t=1 to l sum from k=1 to l((j_t-s,j_k-s)_e),0≤s≤e-1.Theorem 2.4 Let q=ef+1 is an odd prime power, then C_i1^(e), C_i2^(e),...,C_il^(e) is a externaldifference families over GF(q) if and only ifwhere I_s′=sum from 1≤t₁, t₂≤l,t₁≠t₂ to((i_t1-s, i_t2-s)_e),0≤s≤e-1. Theorem 2.5 Let q=ef+1 is an odd prime power, D_j=C_j1^(e)∪C_j2^(e)∪…∪C_jl^e, 1≤j≤h,then D={D₁, D₂,…, D_h}is a external difference families over GF(q) if and only if J′₀=J′₁=…J′_e-1where J′_s=sum from 1≤i≠j≤h sum from 1≤t,k≤l to l(i_t-s,j_k-s))_e,0≤s≤e-1.Theorem 2.9 Let q=2uk+1 is an odd prime power, where k is odd. then D={C₀^(2u),C₁^(2u),…,C_u-1^(2u)}is a (q, k, (k-1)/2)-DDF over GF(q).Theorem 2.10 Let q=2uk+1 is an odd prime power, where u is even. then D={C₀⁽²⁾,C₁⁽²⁾}is a(q,(q-1)/2, (q-1)/2;2)-EDF over GF(q).Theorem 2.11 Let q=2uk+1 is an odd prime power, where uk is odd. then D={D_i:D_i=C_2i^(2u)∪C_2i+1^(2u),0≤i≤u-1}is a (q, 2k, 2k-1)-DDF and a (q, 2k, 2k(u-1); u)-EDF over GF(q).Theorem 2.12 Let q=2uk+1 is an odd prime power, where uk is odd. then D={C_2i^(2u):0≤i≤u-1}is(q,k,(k-1)/2)-DDF and a(q,k,(q-2k-1)/4;u)-EDF over GF(q).Theorem 3.1 Let q=4f+1 is an odd prime power, where f is odd. then D={C₀⁽⁴⁾,C₂⁽⁴⁾}is a (q, f, (f-1)/2)-DDF over GF(q) if and only if q=1+4t².Theorem 3.2 Let q=4f+1 is an odd prime power, where f is even. then D={C₀^(4),C₂4}is an (q, f, f/2; 2)-EDF over GF(q) if and only if q=1+4t².Theorem 3.3 Let q=6f+1 is an odd prime power, where f is odd. then D={C₀⁽⁶⁾,C₁⁽⁶⁾,C₂⁽⁶⁾}is a (q, f, f; 3)-EDF over GF(q) if and only if 2x=a, 2y=b, b=d, q=x²+3y², 4q=a²+3b²=c²+27d².Theorem 3.4 Let q=6f+1 is an odd prime power, where f is odd. then D=C₀⁽⁶⁾∪C₁⁽⁶⁾∪C₂⁽⁶⁾is a (q,3f, (3f-1)/2)difference set over GF(q) if and only if 2y=b, q=x²+3y²,4q=a²+3b²=c²+27d².Theorem 3.5 Let q=6f+1 is an odd prime power, where f is odd. then D₁=C₀⁽⁶⁾∪C₁⁽⁶⁾∪C₂⁽⁶⁾, D₂=C₃⁽⁶⁾∪C₄⁽⁶⁾∪C₅⁽⁶⁾ is a (q, 3f, 3f-1)-DDF over GF(q) and is a (q, 3f, 3f; 2)-EDF overGF(q) if and only if 2y=b,q=x²+3y²,4q=a²+3b²=c²+27d².Theorem 3.6 Let p=8f+1 is an odd prime, where f is even, p is a prime and p=x²+4y²=a²+2b²;x=a=1(mod 4). then D={C₀⁽⁸⁾,C₂⁽⁸⁾,C₄⁽⁸⁾,C₆⁽⁸⁾}is an (p, f, 3f/2; 4)-EDF over GF(p) if and only if x+2a=3.Theorem 3.7 Let p=8f+1 is an odd prime, where f is even, p is a prime and p=x²+4y²=a²+2b²=u²+v²;x≡a≡u≡1 (mod 4), v≡0(mod 8). then D={D_i:D_i=C_2i⁽⁸⁾)∪C_2i+1⁽⁸⁾,0≤i≤3}is an (p, 2f, 2f-1)-DDF and is a (p, 2f, 6f; 4)-EDF over GF(p) if and only if y=-b.The paper is divided into four parts. In chapter one, we present some notions and theircorrelations. In chapter two, we introduce some preliminaries, including definition and propertiesof cyclotomic numbers, the idea of construction and further results of external difference familiesand disjoint difference families over GF(q) with q=2ku+1. In chapter three, we present someresults of difference families via cyclotomic number of special orders. In chapter four, three furtherresearch problems are presented.