Cyclic Difference Families With Block Size Four And Five

Let Z_gv be the set of residue classes of the integers under addition modulo gv and G be asubgroup of order g. A (gv,g,k, 1)-cyclic difference family (or a (gv,g,k, 1)-DF for short) isdefined to be a family S={B₁ ,B₂,……,B_s} of some k-subsets (base blocks) of Z_gv, where Bi ={b_i1,b_i2,……,b_ik},1≤i≤s, such that the differences in S, S = {b_ij-b_it|1≤i≤s,j≠t,1≤j,t≤k}, contain each non-zero element of Z_gv \ G exactly once.The research of cyclic difference family not only play an important role in the theory ofclassical combination design, but also has a good application in communication area. It can beused to construct binary codes in optical code division multiple access which require good auto-correlation and cross-correlation properties. Optical orthogonal codes satisfy the properties. If1≤g≤k(k-1), a (gv,g,k, 1)-DF is a combinatorial characterization of an optimal opticalorthogonal codes of length gv, Hamming weight k, auto-correlation and cross-correlation unity.The existence of cyclic difference family is concerned much more by scholars who are at homeand abroad.In this paper, we obtain the following results by applying the theory of cyclotomic domainto direct and recursive constructions.For any prime p congruent to 7 modulo 12, a (12p,12,4,1)-DF exists.For any prime p congruent to 1 modulo 5, p > 2000 and p≠2081, a (16p,16,5,1)-DFexists.